Abstract
Ultrastrong coupling between light and matter has, in the past decade, transitioned from a theoretical idea to an experimental reality. It is a new regime of quantum light–matter interaction, which goes beyond weak and strong coupling to make the coupling strength comparable to the transition frequencies in the system. The achievement of weak and strong coupling has led to increased control of quantum systems and to applications such as lasers, quantum sensing, and quantum information processing. Here we review the theory of quantum systems with ultrastrong coupling, discussing entangled ground states with virtual excitations, new avenues for nonlinear optics, and connections to several important physical models. We also overview the multitude of experimental setups, including superconducting circuits, organic molecules, semiconductor polaritons, and optomechanical systems, that have now achieved ultrastrong coupling. We conclude by discussing the many potential applications that these achievements enable in physics and chemistry.
Key points

Ultrastrong coupling (USC) can be achieved by coupling many dipoles to light, or by using degrees of freedom whose coupling is not bounded by the smallness of the finestructure constant.

The highest light–matter coupling strengths have been measured in experiments with Landau polaritons in semiconductor systems and in setups with superconducting quantum circuits.

With USC, standard approximations break down, allowing processes that do not conserve the number of excitations in the system, leading to a ground state that contains virtual excitations.

Potential applications of USC include fast and protected quantum information processing, nonlinear optics, modified chemical reactions and the enhancement of various quantum phenomena.

Now that USC has been reached in several systems, it is time to experimentally explore the new phenomena predicted for this regime and to find their useful applications.
INTRODUCTION
The intuitive description of the interaction between light and matter as a series of elementary processes in which a photon is absorbed, emitted or scattered by a distribution of charges essentially hinges on the small value of the fine structure constant \(\alpha \approx \frac{1}{137}\). Because α is the natural dimensionless parameter emerging in a perturbative treatment of quantum electrodynamics, its small value allows most of the quantum dynamics of the electromagnetic field to be described by only taking into account firstorder (absorption, emission) or secondorder (scattering) processes.
Whereas the value of α is fixed by nature, Purcell discovered in 1946 that the strength of the interaction of an emitter with light can be enhanced or suppressed by engineering its electromagnetic environment^{1}. From this crucial observation sprang a whole field of research, today called cavity quantum electrodynamics (QED; Fig. 1), which aims to exploit different kinds of photonic resonators to modulate the coupling of light with matter.
The fundamental and applied importance of controlling the strength of light–matter coupling, g, led to the development of resonators with ever higher quality factors, Q, which are associated with lower energy losses.
In 1983, Haroche and coworkers^{2}, using a collection of Rydberg atoms in a highQ microwave cavity, managed to achieve a coupling strength that exceeded the losses in the system. In this strongcoupling (SC; Fig. 1d) regime it is possible to observe an oscillatory exchange of energy quanta between the matter and the light, called vacuum Rabi oscillations, which takes place at a rate given by g. By contrast, in the weakcoupling (WC; Fig. 1c) regime g is smaller than the losses, and thus the energy is lost from the system before it can be exchanged between the light and the matter.
The SC regime was soon also reached with single atoms coherently interacting with a microwave cavity^{3} and, a few years later, with an optical cavity^{4}. In 1992, the SC regime was demonstrated using quasi2D electronic excitations (Wannier excitons) embedded in a semiconductor optical microcavity^{5}. In this case, the eigenstates of the resulting system are called cavity polaritons. Following these pioneering experiments, cavity QED has been successfully adapted and further developed using artificial atoms, such as quantum dots^{6} and superconducting qubits (circuit QED)^{7}.
In a cavityQED setup, the dimensionless parameter quantifying the interaction is the ratio between the coupling strength g and the bare energy of the excitations. This quantity, the normalized coupling η, is proportional to a positive power of α and its value in the first observations of the SC regime was <10^{−6} for atoms^{4} and 10^{−3} for Wannier excitons in semiconductor microcavities^{5}. Lowestorder perturbation theory is thus perfectly adequate to describe those experiments. The important difference with the WC regime is that, because the coupling is larger than the spectral width of the excitations, degenerate perturbation theory needs to be applied.
It took more than two decades after the observation of SC for the cavityQED community to begin investigating the possibility of accessing a regime with larger η in which higherorder processes, which would hybridize states with different numbers of excitations, become observable. Two main paths were identified to reach such a regime. The first consisted in coupling many dipoles to the same cavity mode (Fig. 1b); as correctly predicted by the Dicke model^{8}, this leads to enhanced coupling that scales with the square root of the number of dipoles. The second path was to use different degrees of freedom, whose coupling is not bounded by the small value of α. An indepth discussion of the scaling of η with α in different physical implementations can be found elsewhere^{9}.
In 2005, following the first path, it was predicted^{10} that this regime, which was named the ultrastrongcoupling (USC; Fig. 1e) regime, could be observed in intersubband polaritons thanks to the large number of electrons involved in the transitions between parallel subbands in a quantum well. In 2009, the USC regime was observed for the first time in a microcavityembedded doped GaAs quantum well, with η = 0.11 (ref.^{11}). Following this initial observation, the value of η = 0.1 has often been taken as a threshold for the USC regime. However, because the intensity of higherorder processes depends continuously on η, the value of 0.1 is just a historical convention, without any deeper physical meaning.
The second path has been followed in experiments with superconducting circuits^{7}, in which USC was observed in 2010, with η = 0.10–0.12 (ref.^{12,13}). In these experiments, it becomes possible to explore the USC of light to a single twolevel system, instead of a collective excitation.
Following these experimental breakthroughs, the interest in USC has blossomed, fostered by the vast phenomenology that has been predicted to be observable in this regime, including modifications of intensity, spectral features and correlations of lightemitting devices with USC^{14,15}, as well as possible modifications of physical or chemical properties of systems ultrastrongly coupled to light^{10,16,17,18,19,20}. This widespread interest led not only to the observation of the USC regime in a large number of physical systems, but also to a steady increase in normalized coupling, whose record is presently η = 1.43 (ref.^{21}).
The achievement of USC can be seen as the beginning of a third chapter in the history of light–matter interaction (Fig. 1). Already the control of this interaction afforded by the Purcell effect in the WC regime had led to several important applications, such as lowthreshold solidstate lasers^{22} and efficient singlephoton and entangledphoton emitters^{23,24}. Cavity QED with individual atoms in the SC regime made it possible to manipulate and control quantum systems, enabling tests of fundamental physics^{25} and applications^{26} such as highprecision measurements^{27} and quantum information processing (QIP)^{28}. As the light–matter coupling strength reaches the USC regime, it starts to become possible to modify the very nature of the light and matter degrees of freedom. This opens new avenues for studying and engineering nonperturbatively coupled light–matter systems, which is likely to lead to novel applications.
In this Review, we gather both theoretical insights and experimental achievements in the field of USC. We begin by discussing various regimes of light–matter coupling in more detail, explaining their similarities and differences, the models used to describe them and their properties. We then examine how USC has been reached in different experimental systems. This is followed by an overview of the defining characteristics of ultrastrong light–matter interaction such as virtual excitations and higherorder processes, which affect how the interaction of an USC system with an environment is treated. We also survey quantum simulations of the USC regime, USC to a continuum instead of a single resonator mode and how ultrastrong light–matter coupling is intimately connected to other areas of physics. We conclude with an outlook for the field, including possible new applications and outstanding challenges.
Light–matter coupling regimes and models
The definitions of the WC, SC and USC regimes compare the light–matter coupling strength g with different parameters (Fig. 1). Whether the coupling is strong or weak depends on whether g is larger than the losses in the system. USC is not SC with larger couplings; its definition does not involve the value of losses but instead compares g with bare energies in the system. It is thus possible for a system to be in the USC regime without having SC if losses are large^{29}. The ratio η that defines USC instead determines whether perturbation theory can be used and to what extent approximations can be made in models for the light–matter interaction.
Models
Some of the most fundamental models describing light–matter interaction, the quantum Rabi, Dicke and Hopfield models, are described in Box 1. These models do not approximate away some terms that are often ignored at low light–matter coupling strengths, but they still rely on various approximations, for example, they assume that the atoms are twolevel systems and that the light is in a single mode. Recently, some of these approximations have been shown to become potentially unsound in the USC regime. Specifically, the inclusion of multiple modes of the photonic resonator has been shown to become a necessity at larger couplings^{30,31,32,33,34}, leading to important changes in the system dynamics even for seemingly safe values of the normalized coupling^{35}. It is also possible to have USC of matter to a continuum of light modes, a scenario discussed below. Moreover, the validity of retaining a single optically active transition in the quantum description of the matter part of the system has recently been demonstrated to be strongly gaugedependent^{36,37}. This remains true even for very anharmonic potentials, in which all the higherlying transitions are spectrally distant from the lowest one, a case in which this kind of approximation is generally considered to be safe^{37}. To solve this issue, the introduction of a generalized minimalcoupling replacement has been proposed^{38}, allowing the gaugeinvariant Hamiltonians on the reduced Hilbert space to be derived.
The light–matter interaction can be divided into two parts (Box 1). It is essential to note that, in contrast to the terms in the first part (weighted by the coupling strength g_{1}), the terms in the second part (weighted by g_{2}) do not conserve \({\widehat{N}}_{{\rm{exc}}}\), the total number of excitations in the system. These latter terms are often referred to as antiresonant or counterrotating. When the light and matter frequencies are close to resonance, these terms can be omitted using the rotatingwave approximation (RWA). In the case of the quantum Rabi model, the RWA simplifies the Hamiltonian to the standard Jaynes–Cummings model^{39} (Box 1). The Jaynes–Cummings model, which has been a workhorse of quantum optics in the WC and SC regimes, conserves \({\widehat{N}}_{{\rm{exc}}}\) and restricts the resulting light–matter dynamics to 2D Hilbert subspaces^{40}. However, the RWA is not justified in the USC regime, when all terms in the light–matter interaction come into play.
Although the quantum Rabi model does not conserve \({\widehat{N}}_{{\rm{exc}}}\), it does conserve the parity \(\widehat{P}={\rm{\exp }}(i\pi {\widehat{N}}_{{\rm{exc}}})\). A generalized quantum Rabi model, which is obtained by replacing the term \(g\widehat{X}{\widehat{\sigma }}_{x}\) by \(g\widehat{X}({\widehat{\sigma }}_{x}{\rm{\cos }}\,\theta +{\widehat{\sigma }}_{z}{\rm{\sin }}\,\theta )\) (with a parameter θ ≠ 0, π) does not conserve even \(\widehat{P}\); this Hamiltonian features in experiments with superconducting circuits^{12,41}. The Jaynes–Cummings model conserves both \({\widehat{N}}_{{\rm{exc}}}\) and \(\widehat{P}\).
An analytical approach to finding the spectrum of the quantum Rabi model was discovered only in 2011 (ref.^{42}), and has since been extended to multiple twolevel systems^{43,44} and bosonic modes^{45}. But it still requires numerical calculations of transcendental (nonanalytical) functions. A particular difficulty is to find exceptional eigenvalues of \({\widehat{H}}_{{\rm{Rabi}}}\) with no definite parity (doubly degenerate)^{42}. In contrast to the quantum Rabi model, the spectrum of the Jaynes–Cummings model is simple and well known^{40}.
The quantum Rabi model can be simulated with the standard Jaynes–Cummings model in experiments using various tricks, as discussed below. Also, the coupling g can be enhanced in various ways, for example, by increasing the number of twolevel systems or cavity fields, or by applying classical (singlephoton) drives to a single twolevel system or a cavity field. Recently, an exponential enhancement of g was predicted with a twophoton drive (squeezing) of the cavity field^{46,47}.
A generalization of the quantum Rabi model to N twolevel systems (which can correspond to a single multilevel system or a large spin) is known as the Dicke model^{8}. Under the RWA, the Dicke model reduces to the Tavis–Cummings model^{48} (Box 1). Another generalized version of the quantum Rabi model, with g_{1} ≠ g_{2}, enables the study of supersymmetry (SUSY), which exists if \({g}_{1}^{2}{g}_{2}^{2}={\omega }_{{\rm{c}}}{\omega }_{{\rm{q}}}\) (that is, when the Bloch–Siegert shift^{49} is zero)^{50}. Note that g_{1} = g_{2} if the Rabi model is derived from first principles.
The Hamiltonian for the Hopfield model is given in Box 1. In this case, the g_{1} terms describe parametric frequency conversion, which conserves the total number of excitations \({\widehat{N}}_{{\rm{exc}}}^{}\), while the g_{2} terms describe parametric amplification, which does not conserve \({\widehat{N}}_{{\rm{exc}}}^{}\). These processes, often studied in quantum optics, are analogous to those described by \(\widehat{H}\)_{int} for the quantum Rabi model. This simplified Hopfield model has been applied to describe experimental data of a 2D electron gas interacting with terahertz cavity photons in the USC regime^{51,52,53}.
Other regimes of light–matter coupling
For the sake of completeness, we mention here three other regimes of light–matter coupling. The first is the deepstrongcoupling (DSC, Fig. 1f) regime, in which η becomes larger than 1 and higherorder perturbative processes are not only observable but can become dominant. Theoretically investigated for the first time in 2010 (ref.^{54}), this regime was finally demonstrated experimentally in 2017 using different physical implementations^{21,41}.
The second, the verystrongcoupling (VSC) regime, is achieved when g becomes comparable to the spacing between the excited levels of the quantum emitter. In this regime, although the number of excitations is conserved and firstorder perturbation gives an adequate description of the system, the coupling is large enough to hybridize different excited states of the emitter, modifying its properties. This regime was initially predicted by Khurgin in 2001 (ref.^{55}), and observed in microcavity polaritons in 2017 (ref.^{56}).
The third is the multimodestrongcoupling (MMSC) regime, in which g exceeds the free spectral range of the resonator that the matter couples to. This regime has recently been reached with superconducting qubits coupled to either microwave photons in a long transmissionline resonator^{31} or phonons in a surfaceacousticwave resonator^{57}.
In the rest of this Review, we largely speak of USC, with the implicit understanding that, according to the value of η and other energy scales, the system under investigation could also be in the WC, SC, VSC, MMSC or DSC regimes.
Properties of systems with USC
As η increases, several properties of coupled light–matter systems change drastically. The lowest energy levels of a light–matter system with a single atom on resonance with a cavity mode as a function of η are plotted in Fig. 2a. Only the quantum Rabi model (Box 1) gives a correct picture of the energy levels for all η; various approximate methods can be used for small or large η. The Jaynes–Cummings model correctly predicts the Rabi splitting (dressed states) between neighbouring pairs of energy levels, but fails when the system enters the USC regime.
Groundstate properties
The difference between the USC and nonUSC regimes is particularly striking for the ground state of a coupled light–matter system (Fig. 2b–e). For small η, the lowest energy state of the system is simply an empty cavity with the atom in its ground state. However, as η grows the coupling makes it increasingly energetically favourable to have atomic and photonic excitations in the ground state. The exact nature of these excitations is discussed in the section on virtual excitations. Here we only note that for very large η, in the DSC regime (Fig. 2e), the ground state of the quantum Rabi model consists of photonic Schrödinger’s cat states entangled with the atom and exhibits nonclassical properties such as squeezing^{16,58}.
The mean number of photons in the ground state starts to increase rapidly when the value of η approaches and passes 1 (Fig. 2b). In the case of many atoms coupled to the light, as described by the Dicke model (Box 1), it is predicted that a quantum phase transition, known as the superradiant phase transition^{59,60,61}, takes place at a critical value of η, separating phases with and without photons in the ground state of the system. However, as explained below, whether or not this phase transition actually occurs depends on whether an additional term, the diamagnetic term, should be included in the Hamiltonian. Furthermore, the recent realization of USC in systems with artificial atoms raises the question of whether the superradiant phase transition occurs when the transition frequencies and coupling strengths of individual atoms differ^{62}. We also note that several works, such as ref.^{63}, have analysed the physics at the critical value of η in the quantum Rabi model instead, in which the thermodynamic limit of infinitely many atoms cannot be realized.
In Fig. 2f we plot the energy levels for the case in which the matter instead consists of many atoms and is described as bosonic collective excitations in the Hopfield model (Box 1). The impact of the diamagnetic term is clearly seen. Figure 2g shows the groundstate population with the diamagnetic term included. Also in this case, the ground state contains virtual light and matter excitations. This ground state can be calculated analytically^{64} for all η; it is a multimode squeezed state for large η.
The diamagnetic term
In the DSC regime, the diamagnetic term (Box 2) can act as a potential barrier for the photonic field, localizing it away from the dipoles, leading to an effective decoupling between the light and matter degrees of freedom^{30}. This means that the Purcell effect, known from the WC regime and thought to increase the rate of spontaneous emission of the qubit with increasing g, actually becomes negligible when g becomes large enough. A similar decoupling can occur if qubit–qubit interactions are added to the Dicke model^{65}. Even in the pure quantum Rabi model, unexpected changes in photon output statistics take place deep in the DSC regime^{66}.
Being a consequence of gauge invariance, the diamagnetic term is required to obtain a consistent theory, including in superconducting systems. Claims have been advanced on the possibility of engineering systems, both dielectric^{67} and superconducting^{68}, in which the diamagnetic term is absent or at least reduced to violate equation B2.4. Those claims have attracted strong criticism^{69,70}. There have also been theoretical proposals showing how the matter could be experimentally settled^{71,72,73}.
This interest in the presence of the diamagnetic term in cavity QED is historically due to its supposed role in stabilizing a system against the occurrence of superradiant phase transitions. It is easy to gain an intuitive understanding of why such a term could stabilize the ground state of the system. The diamagnetic term can be removed from the Hamiltonian by performing a Bogoliubov rotation in the space of the photon operator, at the cost of a renormalization of the cavity frequency: \({\omega }_{{\rm{c}}}\to \sqrt{{\omega }_{{\rm{c}}}^{2}+4{\omega }_{{\rm{c}}}D}\), where D, discussed in Box 2, is a measure of the energy of the diamagnetic term. For the system to undergo a quantum phase transition, the coupling has to be strong enough to push one of the system eigenmodes to zero frequency. The blueshift of the cavity due to the renormalization thus implies that a larger coupling g is required to reach the critical point, but from equation B2.4 this in turn will further blueshift the cavity mode. A careful calculation shows that, at least for the Dicke model, this runaway process leads to a divergent critical value of g if equation B2.4 holds. Notwithstanding this simple argument, a number of works have reported opposite views on the possibility of achieving superradiant phase transitions^{36,69,74,75,76,77,78,79,80,81}. These seemingly contrasting arguments are presently understood to be at least partially due to the role of longrange dipole–dipole interactions, which depend on the specific geometry of the system under consideration and can lead to very different results owing to the choice of gauge and of dynamical variables^{37,76,81}.
The presence in the Hamiltonian of an A^{2} term is not gauge invariant. For example, in the Power–Zienau–Woolley form of the Hamiltonian, in which matter is described by a polarization density P, there is no A^{2} term, but a P^{2} term is present, quadratic in the matter instead of in the photonic field^{77,82}. Finally, it is worth noting that the presence of a squared field term in the Hamiltonian, which assures the stability of matter linearly coupled to a bosonic field, is a feature that is not limited to the interaction with the transverse electromagnetic field. Similar terms, satisfying the equivalent of equation B2.4, have been derived in the case of longitudinal interactions in intersubband polarons^{83}.
Experimental systems with USC
The first experimental demonstration of ultrastrong (η > 0.1) light–matter coupling was reported in 2009 (ref.^{11}). USC has since been achieved in several different systems and at different wavelengths (Fig. 3). In 2017, two experiments even managed to reach DSC (η > 1)^{21,41}. The past decade has seen a rapid increase in values of η (Fig. 3f; Table 1). However, it should be noted that fitting experimental data to theoretical models to extract η can be a subtle and demanding task in the USC and DSC regimes^{35}.
Intersubband polaritons
The USC regime was first predicted^{10} and demonstrated^{11} exploiting intersubband polaritons in microcavityembedded doped quantum wells (Fig. 3a). In these systems, nanoscopic layers of different semiconductors create a confining potential for carriers along the growth direction, which splits electronic bands into discrete parallel subbands. Thanks to the quasiparallel inplane dispersion of the different conduction subbands, all the electrons in the conduction band can be coherently excited, creating narrow collective optical resonances. The coupling of these resonances with transversemagnetic polarized radiation scales with the square root of the total electron density. By modifying the width of the quantum wells, the resonances can be tuned to cover the THz and midinfrared sections of the electromagnetic spectrum.
Intersubbandpolariton systems are usually well described by the Dicke model. This was exploited in ref.^{11}, in which a demonstration of USC with η = 0.11 was obtained by comparing experimental data with best fittings obtained using the Dicke model with and without antiresonant terms.
However, this appealingly simple model is not appropriate for more complex devices. The presence of multiple quasiresonant photonic modes can lead to a physics described by the quantum Rabi model instead^{84}. Moreover, as the width of the quantum wells increases and multiple electron transitions become available, the intuitive picture in terms of singleparticle states is lost. In that case, the electronic transition is better described as a plasmalike mode^{85,86}.
Intersubband polaritons remain a scientifically and technologically interesting system in which to study USC phenomenology thanks to the possibility of nonadiabatically modifying the coupling strength^{87}, which makes it a promising platform for quantum vacuum emission experiments^{88,89}. Moreover, η has been progressively increased in various experiments^{85,90,91,92,93} up to the present value of η = 0.45 (ref.^{86}).
Superconducting circuits
The next experiments to reach USC, in 2010 (ref.^{12,13}), used superconducting quantum circuits. In these systems, electrical circuits with Josephson junctions, operating at GHz frequencies, function as ‘artificial atoms’, acquiring a level structure similar to that of natural atoms when cooled to millikelvin temperatures. These artificial atoms are then coupled to photons in resonators formed by an inductance L and a capacitance C in a lumpedelement circuit or in a transmission line. Superconducting circuits are a powerful platform for exploring atomic physics and quantum optics and for QIP, because their properties (such as the resonance frequencies and coupling strength) can be designed and even tuned in situ^{7}. This has been widely exploited in the SC regime, for example, to engineer quantum states and realize quantum gates.
The superconducting quantum circuit experiments^{12,13,33,41,94,95,96,97,98} are the only ones that have achieved USC with a single (albeit artificial) atom. The reason why superconducting circuits, unlike other systems, do not require collective excitations to reach USC, is that the coupling scales differently with α in these circuits^{9}. In cavity QED, the coupling scales as α^{3/2}. In circuit QED, it scales as either α^{1/2} or α^{−1/2}, depending on the layout of the superconducting circuit^{9}.
The design of the first system to break the DSC barrier^{41} (with η = 1.34) is shown in Fig. 3b. As discussed in more detail below, superconducting quantum circuits are also the only systems in which USC to a continuum^{99,100,101} has been demonstrated, and they have proven to be an excellent platform for quantum simulation of USC^{102,103}.
Landau polaritons
Since 2011, the record for η has almost continuously been held by Landaupolariton systems. In these systems, based on microcavityembedded doped quantum wells under a transverse magnetic field, the USC occurs between a photonic resonator and the collective electronic transitions between continuous Landau levels. In contrast with intersubband polaritons, whose dipole lies along the growth axis, Landau transitions have an inplane dipole and thus couple to transverse electricalpolarized radiation. The very large coupling achievable in these systems is due to an interplay between the degeneracy of Landau levels, the transition dipole, which increases with the index of the highest occupied Landau level, and the relatively small cyclotron frequencies in the THz or GHz range observable in highmobility heterostructures.
Theoretically described for the first time in 2010 (ref.^{51}), Landaupolariton systems with USC were observed shortly afterwards using splitring resonators^{21,104,105,106,107} (Fig. 3c), photoniccrystal cavities^{52,53} and coplanar microresonators^{108}. The present record value of light–matter coupling, η = 1.43, was measured in ref.^{21}.
Landaupolariton systems are a useful platform for investigating USC phenomenology. In ref.^{53}, the polarization selectivity of the Landau transition was used to directly measure the Bloch–Siegert shift due to the antiresonant terms in the Hamiltonian. Furthermore, in ref.^{109}, magnetotransport was used to investigate the nature of the matter excitations participating in the Landaupolariton formation. In ref.^{21}, light–matter decoupling in the DSC regime was reported for the first time, and also exploited to optimize the design of the photonic resonator.
Organic molecules
The USC regime has also been realized at room temperature at a variety of optical frequencies coupling cavity photons (or, in one case, plasmons^{110}) to Frenkel molecular excitons^{14,15,111,112,113,114,115,116}. These systems consist of thin films of organic molecules with giant dipole moments (which make it possible to reach USC) sandwiched between metallic mirrors (Fig. 3d) and present an interesting combination of high coupling strengths and functional capacities. A vacuum Rabi splitting beyond 1 eV, corresponding to η = 0.3, has been reported^{14,115}. Using such high coupling strengths, monolithic organic lightemitting diodes working in the USC regime have been fabricated^{14,15,113,114,116}. These devices exhibit a room temperature dispersionless angleresolved electroluminescence with very narrow emission lines that can be exploited to realize innovative optoelectronic devices.
Optomechanics
The concept of ultrastrong light–matter interaction can be extended to optomechanics. Recently, the USC limit was reached in a setup in which plasmonic picocavities interacted with the vibrational degrees of freedom of individual molecules^{117} (Fig. 3e), achieving η = g/ω_{m} = 0.3 (ω_{m} is the mechanical frequency). The increase in coupling strength is due to the small mode volume of the picocavity, which circumvents the diffraction limit to confine optical light in a volume of a few cubic nanometers.
Another approach to increase the optomechanical coupling strength is to use molecules with high vibrational dipolar strength. This was the approach used in ref.^{32}, in which η = 0.12 was reached. The USC limit has also been approached in circuitoptomechanical systems by using the nonlinearity of a Josephsonjunction qubit to boost η^{118}.
Virtual excitations
A clear difference between USC systems and systems with lower coupling strength is the presence of light and matter excitations in the ground state (Fig. 2). This difference is due to the influence of the counterrotating terms in the Hamiltonian (Box 1). At lower coupling strength, excited states of the system can be ‘dressed states’, superpositions of two states containing both light and matter excitations^{40}. These two states contain the same number of excitations. However, in the USC regime, all excited states are dressed by multiple states containing different numbers of excitations. Much research on USC systems has dealt with understanding whether these excitations dressing the system states (especially the ground state) are real or virtual, how they can be probed or extracted, how they make possible higherorder processes that mirror nonlinear optics^{20} (see also the section on applications) and how they affect the description of input and output for the system (Box 3).
Dressed states and input–output theory
A correct treatment of input–output, decoherence and correlation functions for a USC system requires taking into account that the system operators coupling it to the outside world no longer induce transitions between the bare states of the system (which have fixed numbers of photons and atomic excitations). Instead, the transitions are between the dressed, true eigenstates of the system (which contain contributions from various numbers of photons and atomic excitations)^{16,119,120,121}. Following the development of such a treatment, several interesting properties of USC systems have been revealed. For example, whereas thermal emission of photons is supposed to be bunched (photons tend to be emitted together) and photon emission from a single atom is supposed to be antibunched (photons are emitted one by one), the photons emitted from a thermalized cavity in the USC regime can be antibunched^{122} and a twolevel atom coupled ultrastrongly to a cavity can emit bunched photons^{123}.
A simple way to understand the issue of open quantum systems in the USC regime is to remember that because the Hamiltonian of such a system is nonnumber conserving, its ground state contains a finite population of virtual excitations (Fig. 2). Assuming that the emitted radiation is just proportional to the photon population in the cavity, neglecting to discriminate between real and virtual particles leads to the prediction of unphysical radiation from the ground state^{120,124}. As first shown for confined polaritons^{125}, the quantum operators that correctly describe the emission of an output photon in the USC regime contain contributions from both bare annihilation and bare creation cavityphoton operators.
The resulting input–output relation contains the positivefrequency operator \({\widehat{X}}^{+}={\sum }_{i < j}{X}_{ij} {E}_{i}\rangle \langle {E}_{j} \) instead of the cavitymode annihilation operator a^{120}. Here, \( {E}_{i}\rangle \) are the dressed eigenstates of the USC system, ordered such that E_{j} > E_{i} for j > i. The coefficients X_{ij} are matrix elements between eigenstates. In the simplest case \({X}_{ij}=\langle {E}_{i} {\widehat{a}}+{{\widehat{a}}}^{\dagger } {E}_{j}\rangle \). The operator \({\widehat{X}}^{+}\) can be interpreted as the operator describing the annihilation of physical photons in the interacting system. Analogously, \({\widehat{X}}^{}\equiv {({\widehat{X}}^{+})}^{\dagger }\) corresponds to the creation operator. Interestingly, whereas in the ground state \( {E}_{0}\rangle \) of a system in the USC regime the number of bare photons is nonzero, \(\langle {E}_{0} {a}^{\dagger }a {E}_{0}\rangle \ne 0\) (Fig. 2), the definition of \({\widehat{X}}^{+}\) automatically implies that the number of detectable photons is zero: \(\langle {E}_{0} {\widehat{X}}^{}{\widehat{X}}^{+} {E}_{0}\rangle =0\).
Probing and extracting virtual photons
The photons in the ground state of a system with an atom ultrastrongly coupled to a cavity are not only unable to leave the cavity, they are tightly bound to the atom^{35}. The groundstate photons also cannot be detected by a photoabsorber, even if this absorber is placed inside the cavity, except with very small probability at short timescales set by the time–energy uncertainty^{126}. In light of these properties, the groundstate photons in an USC system are considered virtual. However, even though these virtual photons cannot be absorbed by a detector, there are still ways to probe them. One proposal is to measure the change they produce in the Lamb shift of an ancillary probe qubit coupled to the cavity^{127} (Fig. 4a); another option is to detect the radiation pressure they give rise to if the cavity is an optomechanical system^{128}.
There are also many proposals for how the virtual photons dressing the ultrastrongly coupled ground state \( {E}_{0}\rangle \) (and excited states) can be converted into real ones and extracted from the system. Several of these proposals rely on the rapid modulation of either g or the atomic frequency^{10,88,119,129,130,131,132,133,134,135} (Fig. 4b). The generation of photons through the modulation of a system parameter in this way requires USC, but not SC, highlighting that g is compared with two different parameters in these regimes^{29}. A connection can be made between these schemes and the dynamical Casimir effect, in which vacuum fluctuations are converted into pairs of real photons when a mirror (or another boundary condition) is moved at high speed^{136,137,138,139,140}.
Another way to extract virtual photons is to use additional atomic levels. If only the upper transition in a Ξtype threelevel atom couples ultrastrongly to the cavity, driving the lower transition can switch that USC on and off^{87,141} to create photons^{133}. The virtual photons in the USC part of such a system can also be released through stimulated emission^{142}, which opens up interesting prospects for experimental studies of dressed states in the USC regime^{143} (Fig. 4c). Finally, if the cavity is ultrastrongly coupled to an electronic twolevel system, another way to release photons from \( {E}_{0}\rangle \) is through electroluminescence^{19} (Fig. 4d).
Simulating ultrastrong coupling
Although the USC regime has been reached in several solidstate systems, the experimental effort required to achieve this regime is still considerable. Furthermore, it remains difficult to probe many interesting system properties in these experiments, especially dynamics, for a wide range of parameters. An approach that circumvents these problems is quantum simulation^{144,145}, in which an easytocontrol quantum system is used to simulate the properties of the quantum model of interest. In 2010, such an approach was used to observe^{146} the superradiant phase transition of the Dicke Hamiltonian by placing a Bose–Einstein condensate in an optical cavity and gradually increasing the effective light–matter coupling through an external pump. Another early example is a classical simulation of the dynamics of the parity chains in the quantum Rabi model, realized in an array of femtosecondlaserwritten waveguides in which the waveguide spacing sets the coupling strength and engineered properties of the waveguides set the effective qubit and resonator frequencies^{147,148}.
Several proposals for quantum simulation of USC rely on driving some part of a SC system at two frequencies. Then a rotating frame can be found with renormalized parameters, set by the drives, that can be in the USC regime^{149,150,151,152,153,154,155,156} (drives can also be used to set effective parameters in other ways^{157,158}). In 2017, one such proposal^{150} was implemented in a circuitQED experiment in which two drive tones were applied to a superconducting qubit coupled to a transmissionline resonator^{103} (Fig. 5a). Starting from a bare η <10^{−3}, a simulated η of >0.6 was achieved and the dynamics of population revivals were observed. Recently, the USC was also simulated in a trappedion system^{159} using the proposal of ref.^{152}, and USC between two resonators was simulated in superconducting circuits^{160} following the proposal in ref.^{155}.
However, external continuous drives are not necessary to define a rotating frame that places the system in the USC regime. An ingenious digital quantum simulation can be realized with a system described by the Jaynes–Cummings model (Fig. 5b). By tuning a qubit in and out of resonance with a resonator, and flipping the qubit inbetween, the quantum Rabi Hamiltonian can be simulated^{161,162} (with multiple qubits, this can be straightforwardly extended to simulate the Dicke Hamiltonian^{161,162}). This was the approach taken in another recent circuitQED experiment^{102}, which simulated η up to 1.8 and observed dynamics in this regime, including the evolution of the photonic Schrödinger’s cat states in the ground state of the quantum Rabi model (Fig. 5c), first predicted in ref.^{16}. However, the photons in these simulations are always real, not virtual, like the photons in a physical USC.
Ultrastrong coupling to a continuum
An atom can couple ultrastrongly not only to a single harmonic oscillator, but also to a collection or continuum of them. This constitutes an interesting and, so far, less explored regime of the wellknown spinboson model^{163,164}. To speak of USC to a continuum, the RWA should not be applicable to the interaction terms in the model, which is an extension of the quantum Rabi model (Box 1) to many modes. This roughly corresponds to the relaxation rate Γ of the atom into the continuum being 10% or more of the atomic transition frequency ω_{q}. Note that Γ is determined not only by the coupling g to a single light mode, but also by the density of states J(ω) of the continuum. However, if the continuum is modelled as a 1D array of coupled resonators with the atom coupling to one resonator (Fig. 6a), the η from the singlemode case can still be used to define USC.
After USC to a cavity was first realized a decade ago, several theory proposals showed that superconducting circuits were a suitable platform for USC to a continuum (in this case, an open waveguide on a chip)^{165,166,167}. In 2017, such an experiment succeeded^{99} (Fig. 6b) and more demonstrations have followed^{100,101}. Recently, it has also been shown that USC to a continuum could be simulated in superconducting circuits^{168}, extending the method implemented in ref.^{103} for simulation of USC to a cavity.
Ultrastrong coupling modifies the physics of an atom in a waveguide dramatically compared with when the coupling is low enough for the RWA to be applicable. Similar to the cavity case, the ground state contains a cloud of virtual photons (in many modes) surrounding the atom^{167,169} and the atom transition frequency experiences a strong Lamb shift^{163,164,170,171}. This considerably changes the transmission of photons in the waveguide; the standard scenario, in which the atom reflects single photons on resonance^{172}, no longer holds^{167,170,171,173} (Fig. 6a). Instead, similar to the nonlinearopticslike processes^{20} discussed below, the counterrotating terms allow various frequencyconversion processes^{170,174,175} (Fig. 6c). Other new phenomena include decreasing spontaneous emission rate with increasing coupling^{171} and spontaneous emission of Schrödinger’s cat states^{173}.
Connections to other models
The quantum Rabi Hamiltonian is closely related to a number of other fundamental models and emerging phenomena. These include the Hopfield model, a Jahn–Teller model^{58,165,176,177,178}, a fluctuatinggap model of a disordered Peierls chain^{179} and renormalizationgroup models, such as the spinboson^{166,168} and Kondo models^{166,169,175}. The latter two models can be simulated by superconductingcircuit setups (Fig. 6a). It is counterintuitive, but wellknown, that purely electronic phenomena (such as the Kondo effect) are closely related to strongly dissipative twolevel systems^{163,164}.
Light–matter systems described by a generalized version of the quantum Rabi model (equation B1.1 in Box 1 with g_{1} ≠ g_{2}) enable quantum simulations of supersymmetrical field theories. Specifically, SUSY can be simulated with coupled resonators, each described by the quantum Rabi model and tuned to a SUSY point (or line)^{50}. The quantum Rabi model naturally reveals a certain Bose–Fermi duality, which is the central concept of SUSY. This approach enables topologically protected subspaces to be found, which may help to implement decoherencefree algorithms for QIP. Furthermore, a connection to SUSY breaking has been made for g_{1} = g_{2} when the coupling grows large^{180}. Moreover, dark matter in cosmology may be explained through SUSY, so superconducting quantum circuits in the USC regime could in principle realize darkmatter simulations on a chip.
The quantum Rabi model is also equivalent to a Rashba–Dresselhaus model, describing, for example, a 2D electron gas with spinorbit coupling of Rashba and Dresselhaus types interacting with a perpendicular, constant magnetic field^{50}. This is a fundamental model of condensedmatter physics, which can be realized in many other systems, such as semiconductor heterostructures, quantum wires, quantum dots (confined in parabolic potentials), carbonbased materials, 2D topological insulators, Weyl semimetals and ultracold neutral atoms.
Furthermore, a superconducting quantum circuit with USC has been suggested for demonstrating vacuuminduced symmetry breaking^{181}. This effect is analogous to the Higgs mechanism for the generation of masses of weakforce gauge bosons through gaugesymmetry breaking.
Applications
Why do we need USC when we already have SC? The simplest answer is that USC enables more efficient interactions. For example, the coupling between a single photon and a single emitter results in significant nonlinearity, which has been used in electrooptical devices operating in the SC regime. Increasing η from SC to USC results in better performance of such devices, for example, faster control and response even for shorter lifetimes of the device components. Some quantum effects (including quantum gates) in specific realistic shortlifetime systems cannot be observed below USC.
The list of emerging applications of USC is much longer: QIP, quantum metrology, nonlinear optics, quantum optomechanics, quantum plasmonics, superconductivity, metamaterials, quantum field theory, quantum thermodynamics and even chemistry QED and materials science. Below, we discuss some of these applications in greater detail.
Another question is whether it is possible to predict and observe entirely new phenomena in the USC or DSC regimes. A simple example is the experimental observation of new stable states of matter. These states could be entangled hybrid light–matter ground states in the DSC regime^{41} (Fig. 2), or discrete time crystals, which have been predicted^{182} to exist in systems described by the Dicke model with tunable USC.
Quantum information processing
Cavity and circuitQED systems in the USC regime are especially useful for quantum technologies such as quantum metrology (an example is novel highresolution spectroscopy^{183} using smaller linewidths and improved signaltonoise ratio) and QIP. For QIP, coherent transfer of excitations between light and matter is particularly important. Such transfer can be achieved in the SC regime, but it can be much more efficient in the USC regime. Other QIP applications of USC include extremely fast quantum gate operations^{184,185}, efficient realizations of quantum error correction^{186}, quantum memories^{187,188} (Fig. 7a), protected QIP^{189} (Fig. 7b) and holonomic QIP^{190}. The advantages are not only shorter operation times, but also simpler protocols, in which the natural evolution of a USC system replaces a sequence of quantum gates^{186}. Some of these proposals also exploit the entangled ground states and parity symmetry.
Modifying standard quantum phenomena
Increasing η from SC to USC, various standard quantum phenomena often change drastically. Examples include the Purcell effect^{30}, electromagnetically induced transparency and photon blockade^{66,120}, spontaneous emission spectra^{191}, the Zeno effect^{192,193}, refrigeration in quantum thermodynamics^{194} (Fig. 7c) and photon transfer in coupled cavities^{195}. Such modified effects open up new emerging applications. In particular, lightinduced topology^{196,197,198} and quantum plasmonics^{199} with SC can, in principle, be improved and diversified with USC. Another intriguing development is that USC may help in understanding unconventional superconductivity through studies of lightenhanced (that is, polaritonically enhanced)^{200} and photonmediated^{201} superconductivity.
Higherorder processes and nonlinear optics
The inclusion of the counterrotating terms in the quantum Rabi Hamiltonian also enables prediction of higherorder processes. A prominent example is deterministic nonlinear optics (or vacuumboosted nonlinear optics) with twolevel atoms and (mostly) virtual photons in resonator modes^{20,186,202}. These implementations, in contrast to conventional realizations of various multiwave mixing processes in nonlinear optics, can reach perfect efficiency, need only a minimal number of photons and require only two atomic levels. The counterrotating terms can also be leveraged in USC optomechanics to rapidly construct mechanical quantum states^{203} (Fig. 7d) or observe the dynamical Casimir effect^{140}.
Many nonlinearoptics processes can be described in terms of higherorder perturbation theory involving virtual transitions, in which the system passes from an initial state \( i\rangle \) to the final state \( f\rangle \) via a number of virtual transitions to intermediate states. These virtual transitions do not need to conserve energy, but their sum, the transition from \( i\rangle \) to \( f\rangle \), does. When the light–matter coupling strength increases, the vacuum fluctuations of the electromagnetic field become able to induce such virtual transitions, replacing the role of the intense applied fields in nonlinear optics. In this way, higherorder processes involving counterrotating terms can create an effective coupling between two states of the system (\( i\rangle \) and \( f\rangle \)) with different numbers of excitations^{20}. The strength of the effective coupling g_{eff} approximately scales as gη^{n}, where n is the number of intermediate virtual states visited by the system on the way between \( i\rangle \) and \( \,f\rangle \) (an (n + 1)thorder process).
If the light–matter coupling is sufficiently strong, g_{eff} becomes larger than the relevant decoherence rates in the system (that is, the effective coupling can be termed strong). In this case, the resulting coupling is deterministic, a highly desirable feature for practical applications in quantum technologies. Such a resonant coupling between two states with different numbers of excitations was observed in one of the first USC experiments in 2010 (ref.^{12}).
Subsequent theoretical investigations have shown that these higherorder processes can give rise to intriguing novel cavityQED effects, such as anomalous quantum Rabi oscillations, in which a twolevel atomic transition can coherently emit or absorb photon pairs or triplets^{204,205} (Fig. 7e), or multiple atoms jointly absorb or emit a single photon, each atom taking or providing part of the photon energy^{206}. These deterministic processes enrich the possibilities of using cavityQED for the development of efficient protocols for quantum technologies.
As discussed, superconducting quantum circuits with USC can also be used to simulate other fundamental models and test their predictions, for example, in quantum field theory and solidstate physics. We believe that these applications of the quantum Rabi model to other fundamental models in various branches of physics can stimulate research in all these fields by finding new analogues of condensedmatter effects in quantum–optical systems and vice versa.
Chemistry with ultrastrong coupling
There is increasing interest, theoretical and experimental, in the study of SC and USC cavity QED with molecular ensembles. This may lead to new routes to control chemical bonds and reactions (including dynamics, kinetics and thermodynamics) at the nanoscale level. Such photochemistry of molecular polaritons in optical cavities^{17,207,208} is sometimes referred to as cavity^{209} (or cavitycontrolled^{18}) chemistry, polariton chemistry^{200,210} or QED chemistry^{211,212}. This interest was partially triggered by an experiment that demonstrated the control of the coupling (from WC to USC) between photochromic spiropyran molecules and light in a lowQ metallic cavity^{111}. In this and related experiments^{207}, USC was reached by the collective coupling of many molecules to the cavity mode. To achieve USC (or even only SC) for a single molecule is much more demanding^{213}.
Recent studies show that the excitedstate reactivity of photochemical processes (such as catalysis and photosynthesis) in molecules in nanocavities can be substantially modified by SC and USC^{17,18,210,214}. The reason is that g is comparable to the energies of vibrational and electronic transitions in molecules and their coupling^{17}. In particular, a better control of chemical reactions can be realized via polaron decoupling, induced by SC or USC, of electronic and nuclear degrees of freedom in a molecular ensemble^{18}. Possibilities and limitations of applying USC to change the electronic ground state of a molecular ensemble to control chemical reactions have also been investigated^{17,210}. Some molecular observables depend solely on singlemolecule couplings, whereas others (such as those related to electronically excited states) can also be modified by collective couplings. Moreover, lowbarrier chemical reactions can be affected by the quantum interference of different reaction pathways occurring simultaneously in multiple molecules ultrastrongly coupled to a cavity^{210}.
Some of these studies^{208,209} were based on the quantum Rabi model as in the standard quantumoptical approach. The Dicke model with antiresonant terms was applied to the study of many molecules coupled to a surface plasmon^{210}. Some other works^{212} used a powerful QED densityfunctional formalism of QED chemistry^{211}. This formalism unifies quantum optics and electronicstructure theories by treating a QED system composed of matter and light as a quantum liquid. The original formalism works well for SC, but becomes much less efficient in the USC regime^{212}.
Conclusion and outlook
As we describe in this Review, many intriguing physical effects have been predicted in the USC regime of light–matter interaction. However, related experiments have been limited to increasing the light–matter coupling strength and verifying it by standard transmission measurements. Now that USC has been reached in a broad range of systems, we believe that it is time to experimentally explore the new interesting phenomena specific to USC and, finally, to find their useful applications. A few decades ago, cavity QED in the WC and SC regimes was following the same route, which led to important applications in modern quantum technologies. We believe that USC applications also have the potential to make a profound impact.
Change history
26 February 2019
The following changes have been made to the original article: in the lowerright panel of Fig. 1, opoelectrics has been corrected to optoelectronics; in the Box 1 footnote, rotating waveapproximation has been corrected to rotatingwave approximation; in equation B1.3, an operator symbol has been added to the last term; and in the third paragraph of Box 2, j→n⟩ was changed to j⟩→n⟩. This has been corrected in the HTML and PDF versions of the article.
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Acknowledgements
Numerical simulations were performed using QuTiP^{215}. The authors thank Y.X. Liu, X. Gu, O. Di Stefano and A. Ridolfo for useful discussions. The authors also thank A. Settineri, M. Cirio and S. Ahmed for technical assistance with some of the figures. A.F.K. acknowledges partial support from a JSPS Postdoctoral Fellowship for Overseas Researchers (P15750). A.M. and F.N. acknowledge support from the Sir John Templeton Foundation. S.D.L. acknowledges support from a Royal Society research fellowship and thanks F.N. for his hospitality at RIKEN during the course of this work. F.N. also acknowledges support from the MURI Center for Dynamic MagnetoOptics via the Air Force Office of Scientific Research (AFOSR) award No. FA95501410040, the Army Research Office (ARO) under grant No. W911NF1810358, the Asian Office of Aerospace Research and Development (AOARD) grant No. FA23861814045, the Japan Science and Technology Agency (JST) through the QLEAP program, the ImPACT program, and CREST grant No. JPMJCR1676, the Japan Society for the Promotion of Science (JSPS) through the JSPSRFBR grant No. 175250023 and the JSPSFWO Grant No. VS.059.18N, and the RIKENAIST Challenge Research Fund.
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