Gauge non-invariance due to material truncation has recently been explored in a number of contexts in strong-coupling QED. We show that the approach proposed recently in Nature Physics 15, 803 (2019) rests on an incorrect mathematical assertion and so does not resolve gauge non-invariance. It produces new two-level models that are not equivalent in different gauges. The new Coulomb-gauge model is inaccurate for the regimes considered in Nature Physics 15, 803 (2019), for which the multipolar-gauge quantum Rabi model is accurate. The models analysed in Nature Physics 15, 803 (2019) do not result from the argument provided in the main text, but instead from truncation within the multipolar-gauge followed by the application of a truncated phase-invariance principle. More generally, this principle can be applied following truncation in any gauge and it yields an equivalence class of truncated models within the gauge chosen. Equivalence classes belonging to different gauges are not equivalent and the truncated phase-invariance principle does not provide an argument to prefer a particular class. In general, the optimal class depends on the physical situation, including the observables, the parameter regime, and the number of field modes being considered. We also emphasise that gauge-ambiguities are not synonymous with gauge non-invariance due to approximations. Independent of material truncation subsystem predictions can be vastly different depending on the theoretical definitions of the subsystems, which are controlled by the gauge choice. This constitutes an example of vector-space relativity that in no way contradicts gauge-invariance. However, within ultrastrong-coupling regimes this relativity can no longer be ignored.
We investigate optical response of a linear waveguide quantum electrodynamics (QED) system, namely, an optical cavity coupled to a waveguide. Our analysis is based on exact diagonalization of the overall Hamiltonian and is therefore rigorous even in the ultrastrong coupling regime of waveguide QED. Owing to the counter-rotating terms in the cavity-waveguide coupling, the motion of cavity amplitude in the phase space is elliptical in general. Such elliptical motion becomes remarkable in the ultrastrong coupling regime due to the large Lamb shift comparable to the bare cavity frequency. We also …
We study ultrastrong-coupling quantum-phase-transition phenomena in a few-qubit system. In the one-qubit case, three second-order transitions occur and the Goldstone mode emerges under the condition of ultrastrong-coupling strength. Moreover, a first-order phase transition occurs between two different superradiant phases. In the two-qubit case, a two-qubit Hamiltonian with qubit-qubit interactions is analyzed fully quantum mechanically. We show that the quantum phase transition is inhibited even in the ultrastrong-coupling regime in this model. In addition, in the three-qubit model, the superr…
We discuss the properties of bound states in finite-bandwidth waveguide QED beyond the Rotating Wave Approximation or excitation number conserving light-matter coupling models. Therefore, we extend the \emph{standard} calculations to a broader range of light-matter strengths, in particular, in the so-called ultrastrong coupling regime. We do this using the Polaron technique. Our main results are as follows. We compute the spontaneous emission rate, which is renormalized as compared to the Fermi Golden Rule formula. We generalise the existence criteria for bound states, their properties and the…
In ultra- and deep-strong cavity quantum electrodynamics (QED) systems, many intriguing phenomena that do not conserve the excitation number are expected to occur. In this study, we theoretically analyze the optical response of an ultrastrong cavity-QED system in which an atom is coupled to the fundamental and third harmonic modes of a cavity, and report the possibility of deterministic three-photon down-conversion of itinerant photons upon reflection at the cavity. In the conventional parametric down-conversion, a strong input field is needed because of the smallness of the transition matrix …
A quantum dipole interacting with an optical cavity is one of the key models in cavity quantum electrodynamics (cavity-QED). To treat this system theoretically, the typical approach is to truncate the dipole to two levels. However, it has been shown that in the ultrastrong-coupling regime, this truncation naively destroys gauge invariance. By truncating in a manner consistent with the gauge principle, we introduce master equations {for open systems} to compute gauge-invariant emission spectra, photon flux rates, and quantum correlation functions which show significant disagreement with previou…
Higher-order transitions can occur in the ultrastrong-coupling regime of circuit QED through virtual processes governed by the counter-rotating interactions. We propose a feasible way to probe higher-order transitions through the scattering of propagating microwave photons incident on the hybrid qubit-cavity system. The lineshapes in the scattering spectra can indicate the coherent interaction between the qubits and the cavity, and the higher-order transitions can be identified in the population spectra. We further find that if the coupling strengths between the two qubits and the cavity are t…
We propose a quantum simulation of a two-level atom coupled to a single mode of the electromagnetic field in the ultrastrong-coupling regime based upon resonant Raman transitions in an atom interacting with a high finesse optical cavity mode. We show by numerical simulation the possibility of realizing the scheme with a single rubidium atom, in which two hyperfine ground states make up the effective two-level system, and for cavity QED parameters that should be achievable with, for example, microtoroidal whispering-gallery-mode resonators. Our system also enables simulation of a generalized mo…
Cavity and circuit QED study light-matter interaction at its most fundamental level. Yet, this interaction is most often neglected when considering the coupling of this system with an environment. In this paper, we show how this simplification, which leads to the standard quantum optics master equation, is at the root of unphysical effects. Including qubit relaxation and dephasing, and cavity relaxation, we derive a master equation that takes into account the qubit-resonator coupling. Special attention is given to the ultrastrong coupling regime, where the failure of the quantum optical master…
Quantum optics with giant emitters has shown a new route for the observation and manipulation of non-Markovian properties in waveguide-QED. In this paper we extend the theory of giant atoms, hitherto restricted to the perturbative light-matter regime, to deal with the ultrastrong coupling regime. Using static and dynamical polaron methods we address the low energy subspace of a giant atom coupled to an Ohmic waveguide beyond the standard rotating wave approximation. We analyze the equilibrium properties of the system by computing the atomic frequency renormalization as a function of the coupli…
We theoretically study a circuit quantum electrodynamics (QED) architecture with superconducting flux qubits. The qubit is coupled to the transmission line resonator by an ac current originating from the current mode of the resonator. Ultrastrong coupling can be obtained by varying the capacitance between the qubit and the resonator. We propose a scalable design where the two-qubit coupling can be achieved.
When an atom is strongly coupled to a cavity, the two systems can exchange a single photon through a coherent Rabi oscillation. This process enables precise quantum-state engineering and manipulation of atoms and photons in a cavity, which play a central role in quantum information and measurement. Recently, a new regime of cavity QED has been reached experimentally where the strength of the interaction between light and artificial atoms (qubits) becomes comparable to the atomic transition frequency or the resonance frequency of the cavity mode. Here we show that this regime can strongly modif…
The giant atom system is a new paradigm in quantum optics, in which the traditional dipole approximation is not available. In this paper, we construct an artificial giant atom model by coupling a superconducting circuits to a transmission line by two coupling points. In the ultrastrong coupling regime, we show that the Lamb shift of the giant atom, which is induced by the non-negligible counter-rotating atom-waveguide coupling term, will modify its dissipation process. Furthermore, we investigate quantum Zeno and anti-Zeno effect where the size of the giant atom serves as a sensitive controlle…
Starting from the paradigmatic spin-boson model (SBM), we investigate the static and dynamical properties of a system of two distant two-level emitters coupled to a one-dimensional Ohmic waveguide beyond the rotating wave approximation. Employing static and dynamical polaron Ansätze we study the effects of finite separation distance on the behavior of the photon-mediated Ising-like interaction, qubit frequency renormalization, ground-state magnetization, and entanglement entropy of the two-qubit system. Based on previous works we derive an effective approximate Hamiltonian for the two-impurity…
In quantum field theory, bare particles are dressed by a cloud of virtual particles to form physical particles. The virtual particles affect properties such as the mass and charge of the physical particles, and it is only these modified properties that can be measured in experiments, not the properties of the bare particles. The influence of virtual particles is prominent in the ultrastrong-coupling regime of cavity quantum electrodynamics (QED), which has recently been realized in several condensed-matter systems. In some of these systems, the effective interaction between atom-like transitio…
We propose a superconducting circuit platform for simulating spin-1 models. To this purpose we consider a chain of N ultrastrongly coupled qubit-resonator systems interacting through a grounded SQUID. The anharmonic spectrum of the qubit-resonator system and the selection rules imposed by the global parity symmetry allow us to activate well controlled two-body quantum gates via AC-pulses applied to the SQUID. We show that our proposal has the same simulation time for any number of spin-1 interacting particles. This scheme may be implemented within the state-of-the-art circuit QED in the ultras…
We discuss the equilibrium and out of equilibrium dynamics of cavity QED in presence of dissipation beyond the standard perturbative treatment of losses. Using the dynamical polaron \emph{ansatz} and Matrix Product State simulations, we discuss the case where both light-matter $g$-coupling and system-bath interaction are in the ultrastrong coupling regime. We provide a critical $g$ for the onset of Rabi oscillations. Besides, we demonstrate that the qubit is \emph{dressed} by the cavity and dissipation. That such dressing governs the dynamics and, thus, it can be measured. Finally, we sketch a…
We propose a method to get experimental access to the physics of the ultrastrong (USC) and deep strong (DSC) coupling regimes of light-matter interaction through the quantum simulation of their dynamics in standard circuit QED. The method makes use of a two-tone driving scheme, using state-of-the-art circuit-QED technology, and can be easily extended to general cavity-QED setups. We provide examples of USC/DSC quantum effects that would be otherwise unaccessible.
We show that spontaneous Raman scattering of incident radiation can be observed in cavity-QED systems without external enhancement or coupling to any vibrational degree of freedom. Raman scattering processes can be evidenced as resonances in the emission spectrum, which become clearly visible as the cavity-QED system approaches the ultrastrong coupling regime. We provide a quantum mechanical description of the effect, and show that ultrastrong light-matter coupling is a necessary condition for the observation of Raman scattering. This effect, and its strong sensitivity to the system parameters…
Ultrastrong-coupling between two-level systems and radiation is important for both fundamental and applied quantum electrodynamics (QED). Such regimes are identified by the breakdown of the rotating-wave approximation, which applied to the quantum Rabi model (QRM) yields the apparently less fundamental Jaynes-Cummings model (JCM). We show that when truncating the material system to two levels, each gauge gives a different description whose predictions vary significantly for ultrastrong-coupling. QRMs are obtained through specific gauge choices, but so too is a JCM without needing the rotating-…
We propose a new method for frequency conversion of photons which is both versatile and deterministic. We show that a system with two resonators ultrastrongly coupled to a single qubit can be used to realize both single- and multiphoton frequency-conversion processes. The conversion can be exquisitely controlled by tuning the qubit frequency to bring the desired frequency-conversion transitions on or off resonance. Considering recent experimental advances in ultrastrong coupling for circuit QED and other systems, we believe that our scheme can be implemented using available technology.
Gauge invariance is the cornerstone of modern quantum field theory. Recently, it has been shown that the quantum Rabi model, describing the dipolar coupling between a two-level atom and a quantized electromagnetic field, violates this principle. This widely used model describes a plethora of quantum systems and physical processes under different interaction regimes. In the ultrastrong coupling regime, it provides predictions which drastically depend on the chosen gauge. This failure is attributed to the finite-level truncation of the matter system. We show that a careful application of the gau…
We present an experimentally feasible scheme to implement holonomic quantum computation in the ultrastrong-coupling regime of light-matter interaction. The large anharmonicity and the Z2 symmetry of the quantum Rabi model allow us to build an effective three-level Λ-structured artificial atom for quantum computation. The proposed physical implementation includes two gradiometric flux qubits and two microwave resonators where single-qubit gates are realized by a two-tone driving on one physical qubit, and a two-qubit gate is achieved with a time-dependent coupling between the field quadratures …
We investigate theoretically the quantum vacuum properties of a chain of $N$ superconducting Josephson atoms inductively coupled to a transmission line resonator. We derive the quantum field Hamiltonian for such circuit-QED system, showing that, due to the type and strength of the interaction, a quantum phase transition can occur with a twice degenerate quantum vacuum above a critical coupling. In the finite-size case, the degeneracy is lifted, with an energy splitting decreasing exponentially with increasing values of $g^2 N^2$, where $g$ is the dimensionless vacuum Rabi coupling per artifici…
We study a circuit QED setup where multiple superconducting qubits are ultrastrongly coupled to a single radio-frequency resonator. In this extreme parameter regime of cavity QED the dynamics of the electromagnetic mode is very slow compared to all other relevant timescales and can be described as an effective particle moving in an adiabatic energy landscape defined by the qubits. The focus of this work is placed on settings with two or multiple qubits, where different types of symmetry-breaking transitions in the ground- and excited-state potentials can occur. Specifically, we show how the ch…
We study the squeezing of output quadratures of an electro-magnetic field escaping from a resonator coupled to a general quantum system with arbitrary interaction strengths. The generalized theoretical analysis of output squeezing proposed here is valid for all the interaction regimes of cavity-quantum electrodynamics: from the weak to the strong, ultrastrong, and deep coupling regimes. For coupling rates comparable or larger then the cavity resonance frequency, the standard input-output theory for optical cavities fails to calculate the correct output field-quadratures and predicts a non-negl…
We investigate theoretically the performance of two-qubit resonant gates in the crossover from the strong to the ultrastrong coupling (USC) regime of light-matter interaction in circuit QED. Two controlled-PHASE gate schemes---that works well within the rotating wave-approximation (RWA)---are analyzed while taking into account the effects of counter-rotating terms appearing in the Hamiltonian. Our numerical results show that the fidelity of the gate operation is above 96 % when the ratio between the coupling strength and the resonator frequency, g/ω_r, is of about 10 %. Novel schemes are requi…
We study the Zeno and anti-Zeno effects in a superconducting qubit interacting strongly and ultrastrongly with a microwave resonator. Using a model of a frequently measured two-level system interacting with a quantized mode, we show different behaviors and total control of the Zeno times depending on whether the rotating-wave approximation can be applied in the Jaynes-Cummings model, or not. We exemplify showing the strong dependence of our results with the properties of the initial field states and suggest applications for quantum tomography.
We revisit the derivation of Rabi- and Dicke-type models, which are commonly used for the study of quantum light-matter interactions in cavity and circuit QED. We demonstrate that the validity of the two-level approximation, which is an essential step in this derivation, depends explicitly on the choice of gauge once the system enters the ultrastrong coupling regime. In particular, while in the electric dipole gauge the two-level approximation can be performed as long as the Rabi frequency remains much smaller than the energies of all higher-lying levels, it can dramatically fail in the Coulom…
We report on ultrastrong coupling between a superconducting flux qubit and a resonant mode of a system comprised of two superconducting coplanar stripline resonators coupled galvanically to the qubit. With a coupling strength as high as 17% of the mode frequency, exceeding that of previous circuit quantum electrodynamics experiments, we observe a pronounced Bloch-Siegert shift. The spectroscopic response of our multimode system reveals a clear breakdown of the Jaynes-Cummings model. In contrast to earlier experiments, the high coupling strength is achieved without making use of an additional i…
We investigate theoretically the dynamical behavior of a qubit obtained with the two ground eigenstates of an ultrastrong coupling circuit-QED system consisting of a finite number of Josephson fluxonium atoms inductively coupled to a transmission line resonator. We show an universal set of quantum gates by using multiple transmission line resonators (each resonator represents a single qubit). We discuss the intrinsic 'anisotropic' nature of noise sources for fluxonium artificial atoms. Through a master equation treatment with colored noise and manylevel dynamics, we prove that, for a general c…
In cavity quantum electrodynamics (QED), light-matter interaction is probed at its most fundamental level, where individual atoms are coupled to single photons stored in three-dimensional cavities. This unique possibility to experimentally explore the foundations of quantum physics has greatly evolved with the advent of circuit QED, where on-chip superconducting qubits and oscillators play the roles of two-level atoms and cavities, respectively. In the strong coupling limit, atom and cavity can exchange a photon frequently before coherence is lost. This important regime has been reached both i…
We theoretically study a circuit QED architecture based on a superconducting flux qubit directly coupled to the center conductor of a coplanar waveguide transmission-line resonator. As already shown experimentally [Abdumalikov et al. Phys. Rev. B 78, 180502 (2008)], the strong coupling regime of cavity QED can readily be achieved by optimizing the local inductance of the resonator in the vicinity of the qubit. In addition to yielding stronger coupling with respect to other proposals for flux qubit based circuit QED, this approach leads to a qubit-resonator coupling strength g which does not sc…
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 sum…
Superconducting quantum circuits possess the ingredients for quantum information processing and for developing on-chip microwave quantum optics. From the initial manipulation of few-level superconducting systems (qubits) to their strong coupling to microwave resonators, the time has come to consider the generation and characterization of propagating quantum microwaves. In this paper, we design a key ingredient that will prove essential in the general frame: a swtichable coupling between qubit(s) and transmission line(s) that can work in the ultrastrong coupling regime, where the coupling stren…
We propose a reliable scheme to realize the ultrastrong Jaynes-Cummings (JC) model by simultaneously modulating the resonance frequencies of the two-level system and the bosonic mode in the ultrastrong quantum Rabi model. We find that in both the high- and low-frequency modulation regimes, the counter-rotating terms can be completely suppressed without reducing the coupling strength of the rotating-wave terms, and hence the ultrastrong JC Hamiltonian is achieved. The ultrastrong JC interaction can not only be used to implement ultrafast quantum operations, but also will open up a new route to …
We study effective light-matter interactions in a circuit QED system consisting of a single $LC$ resonator, which is coupled symmetrically to multiple superconducting qubits. Starting from a minimal circuit model, we demonstrate that in addition to the usual collective qubit-photon coupling the resulting Hamiltonian contains direct qubit-qubit interactions, which have a drastic effect on the ground and excited state properties of such circuits in the ultrastrong coupling regime. In contrast to a superradiant phase transition expected from the standard Dicke model, we find an opposite mechanism…
We study thermal emission of a cavity quantum electrodynamic system in the ultrastrong-coupling regime where the atom-cavity coupling rate becomes comparable the cavity resonance frequency. In this regime, the standard descriptions of photodetection and dissipation fail. Following an approach that was recently put forward by Ridolfo et al.[arXiv:1206.0944], we are able to calculate the emission of systems with arbitrary strength of light matter interaction, by expressing the electric field operator in the cavity-emitter dressed basis. Here we present thermal photoluminescence spectra, calculat…
The interaction among the components of a hybrid quantum system is often neglected when considering the coupling of these components to an environment. However, if the interaction strength is large, this approximation leads to unphysical predictions, as has been shown for cavity-QED and optomechanical systems in the ultrastrong-coupling regime. To deal with these cases, master equations with dissipators retaining the interaction between these components have been derived for the quantum Rabi model and for the standard optomechanical Hamiltonian. In this article, we go beyond these previous der…
Photon antibunching in the light scattered by single quantum emitters is one of the hallmarks of quantum optics, providing an unequivocal demonstration of the quantized nature of the electromagnetic field. Antibunching can be intuitively understood by the need for a two-level system lying in its lower state after emitting a photon to be re-excited into the upper one before a second emission can take place. Here we show that such a picture breaks down in the ultrastrong light-matter coupling regime, when the coupling strength becomes comparable to the bare emitter frequency. Specialising to the…
We report on spectra of circuit-quantum-electrodynamics (QED) systems in an intermediate regime that lies between the ultrastrong and deep-strong-coupling regimes, which have been reported previously in the literature. Our experimental results, along with numerical simulations, demonstrate that as the coupling strength increases, the spectrum of a circuit-QED system undergoes multiple qualitative transformations, such that several coupling regimes are identified, each with its own unique spectral features. The different spectral transformations can be related to crossings between energy level …
In the standard theory of cavity quantum electrodynamics (QED), coupling between photons inside and outside a cavity (cavity system and photonic reservoir) is given conserving the total number of photons. However, when the cavity photons (ultra)strongly interact with atoms or excitations in matters, the system-reservoir coupling must be determined from a more fundamental viewpoint. Based on the Maxwell's boundary conditions in the QED theory for dielectric media, we derive the quantum Langevin equation and input-output relation, in which the total number of polaritons (not photons) inside the …
We develop a rigorous theoretical framework for interaction-induced phenomena in the waveguide quantum electrodynamics (QED) driven by mechanical oscillations of the qubits. Specifically, we predict that the simplest set-up of two qubits, harmonically trapped over an optical waveguide, enables the ultrastrong coupling regime of the quantum optomechanical interaction. Moreover, the combination of the inherent open nature of the system and the strong optomechanical coupling leads to emerging parity-time (\PT) symmetry, quite unexpected for a purely quantum system without artificially engineered …
Ultrastrongly coupled quantum hardware may increase the speed of quantum state processing in distributed architectures, allowing to approach fault-tolerant threshold. We show that circuit QED architectures in the ultrastrong coupling regime, which has been recently demonstrated with superconductors, may show substantial speedup for a class of adiabatic protocols resilient to the main source of errors, namely the interplay of dynamical Casimir effect and cavity losses.
A cavity quantum electrodynamical (QED) system beyond the strong-coupling regime is expected to exhibit intriguing quantum phenomena. Here we report a direct measurement of the photon-dressed qubit transition frequencies up to four photons by harnessing the same type of state transitions in an ultrastrongly coupled circuit-QED system realized by inductively coupling a superconducting flux qubit to a coplanar-waveguide resonator. This demonstrates a convincing observation of the photon-dressed Bloch-Siegert shift in the ultrastrongly coupled quantum system. Moreover, our results show that the p…
Exotic quantum vacuum phenomena are predicted in cavity quantum electrodynamics (QED) systems with ultrastrong light-matter interactions. Their ground states are predicted to be vacuum squeezed states with suppressed quantum fluctuations. The source of such phenomena are antiresonant terms in the Hamiltonian, yet antiresonant interactions are typically negligible compared to resonant interactions in light-matter systems. We report an unusual coupled matter-matter system of magnons that can simulate a unique cavity QED Hamiltonian with coupling strengths that are easily tunable into the ultrast…
Plasmonic dimer cavities can induce extreme electric-field hot spots that allow one to access ultrastrong coupling regimes using Raman-type spectroscopy on single vibrating molecules. Using a generalized master equation, we study resonant Raman scattering in the strong coupling regime of cavity-QED, when also in the vibrational ultrastrong coupling regime, leading to "phonon-dressed polaritons". The master equation rigorously includes spectral baths for the cavity and vibrational degrees of freedom, as well as a pure dephasing bath for the resonant two-level system, which play a significant ro…
Ultrastrong coupling may allow faster operations for the development of quantum technologies at the expenses of increased sensitivity to new kind of intrinsic errors. We study state transfer in superconducting circuit QED architectures in the ultrastrong coupling regime. Using optimal control methods we find a protocol resilient to the main source of errors, coming from the interplay of the dynamical Casimir effect with cavity losses.
We analyze a multiqubit circuit QED system in the regime where the qubit-photon coupling dominates over the system's bare energy scales. Under such conditions a manifold of low-energy states with a high degree of entanglement emerges. Here we describe a time-dependent protocol for extracting these quantum correlations and converting them into well-defined multipartite entangled states of noninteracting qubits. Based on a combination of various ultrastrong-coupling effects, the protocol can be operated in a fast and robust manner, while still being consistent with experimental constraints on sw…
We study the interaction of a two-level atom with a single-mode nonlinear electromagnetic resonator, considering coupling strengths ranging from zero to the so-called deep strong coupling regime. When the qubit-resonator coupling is very strong, the standard Kerr model for the resonator becomes questionable. Moreover, recently, it has been shown that extra care is needed when constructing gauge-independent theories in the presence of approximations as the truncation of the Hilbert space of the matter system. Such a truncation can ruin gauge invariance leading to non-physical results, especiall…
We investigate theoretically how the spectroscopy of an ancillary qubit can probe cavity (circuit) QED ground states containing photons. We consider three classes of systems (Dicke, Tavis-Cummings and Hopfield-like models), where non-trivial vacua are the result of ultrastrong coupling between N two-level systems and a single-mode bosonic field. An ancillary qubit detuned with respect to the boson frequency is shown to reveal distinct spectral signatures depending on the type of vacua. In particular, the Lamb shift of the ancilla is sensitive to both ground state photon population and correlat…
The ultrastrongly coupling (USC) system has very important research significance in quantum simulation and quantum computing. In this paper, the ultranarrow spectrum of a circuit QED system with two qubits ultrastrongly coupled to a single-mode cavity is studied. In the regime of USC, the JC model breaks down and the counter-rotating terms in the quantum Rabi Hamiltonian leads to the level anti-crossing in the energy spectrum. Choosing a single-photon driving field at the point of avoided-level crossing, we can get an equivalent four-level dressed state model, in which the dissipation of the t…
We introduce a class of quantum optical Hamiltonian characterized by three-body couplings, and propose a circuit-QED scheme based on state-of-the-art technology that implements the considered model. Unlike two-body light-matter interactions, this three-body coupling Hamiltonian is exclusively composed of terms which do not conserve the particle number. We explore the three-body ultrastrong coupling regime, showing the emergence of a superradiant phase transition which is of first order, is characterized by the breaking of a $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry and has a strongly non-Gaus…
We present a circuit-QED scheme which allows to reach the ultrastrong coupling regime of a nondipolar interaction between a single qubit and a quantum resonator. We show that the system Hamiltonian is well approximated by a two-photon quantum Rabi model and propose a simple scattering experiment to probe its fundamental properties. In particular, we identify a driving scheme that reveals the change in selection rules characterizing the breakdown of the rotating-wave approximation and the transition from strong to ultrastrong two-photon interactions. Finally, we show that a frequency crowding i…
