**Polariton Panorama **

**D.N. Basov ^{1}, A. Asenjo-Garcia^{1}, P.J. Schuck^{2}, X.Y. Zhu^{3}, Angel Rubio^{4,5}**

*1.Department of Physics, Columbia University, New York, NY 10027, USA.*

* 2.Department of Mechanical Engineering, Columbia University, New York, NY, USA *

*3.Department of Chemistry, Columbia University, New York, NY 10027, USA. *

*4.Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany*

*5.Center for Computational Quantum Physics (CCQ), Flatiron Institute, 162 Fifth Avenue, New York, New York 10010, USA*

**Abstract: **In this brief review, we summarize and elaborate on some of the nomenclature of polaritonic phenomena and systems as they appear in the literature on quantum materials and quantum optics. This summary includes at least sixty different types of polaritonic light-matter dressing effects unraveling a broad panorama of the physics and technology of polaritons.

Polaritons are commonly described as light-matter hybrid quasiparticles. Polaritons inherit their attributes from both their light and matter constituents. More rigorously, a polariton is a quantum mechanical superposition of a photon with a matter excitation, the latter being a collective mode in solids and superconducting circuits or an electron in atoms, molecules or even superconducting qubits. As such, the notion of polaritons is a unifying universal concept between the fields of quantum materials and quantum optics/electrodynamics. Until fairly recently, these subfields of contemporary physics evolved largely independently of each other. Among the unintended consequences of these divisions is the ambiguity in polaritonic terminology with the same terms used markedly differently in quantum materials (QM) and cavity quantum electrodynamics (QED) in atomic systems. Here, we attempt to summarize (in alphabetical order) some of the polaritonic nomenclature in the two subfields. We hope this summary will help readers to navigate through the vast literature in both of these fields (Refs. 1–496). Apart from its utilitarian role, this summary presents a broad panorama of the physics and technology of polaritons transcending the specifics of particular polaritonic platforms (Boxes 1–2). We invite readers to consult with reviews covering many important aspects of the physics of polaritons in quantum materials [1]–[3], atomic and molecular systems [4] and in circuit QED [5], [6] as well as general reviews of the closely related topic of strong light-matter interaction [7]–[11].

##### Box 1 - Cavity quantum electrodynamics and cavity polaritons

Box 1: Cavity quantum electrodynamics and cavity polaritons. In cavity quantum electrodynamics (QED), the spontaneous emission of atoms, molecules and solids is governed not only by the properties of the emitter *per se* but is also controlled by its local electromagnetic environment. Optical cavities assembled from two parallel mirrors have long been used to confine light, to enhance light-matter interaction and to promote lasing [19]. The probability of interaction between light and matter is enhanced by the number of bounces the photon makes between the mirrors before leaving the cavity, which is conventionally quantified by the cavity finesse *F*. Cavities with high quality factors promote extremely efficient light matter couplings. In the strong-coupling regime (where the coherent interactions between the matter excitation and the cavity mode overcome the dissipation, i.e., when the vacuum Rabi splitting is much larger than the linewidth), the atomic or material excitation hybridizes with the photonic mode and produces a cavity polariton. The minimum separation upper polariton branch and lower polariton branch E_{UPB} −E_{LPB} in Panel H is commonly referred to the normal-mode splitting in analogy to the Rabi splitting of a single-atom cavity system [20] (also Fig.4). Rabi splitting can reach fractions of eV in quantum materials and can exceed 1 eV in molecules [21], [22]. Strong coupling leads to photon blockade, where the presence of a photon in a cavity blocks a second one from coming in the study by Tian and Carmichael [23] and Imamoğlu et al. [24]. See also *microcavity polaritons**.*

##### Figure 1 - Collective modes of a 2D superconductor

Figure 1: Schematic representation in the frequency–momentum plane of the collective modes that may appear in the electrodynamical response of a two-dimensional (2D) superconductor. The blue area shows the low-energy and long-wavelength region, where weakly damped collective modes may be observed. Anticrossing between the plasmon and Higgs mode and the Bardasis–Schrieffer (BaSh) mode is not shown here. Here, c is the speed of light, vF is the Fermi velocity, and Df is the normal-state diffusion coefficient. Adapted from a study by Sun et al. [13].

##### Box 2 - Panorama of cavities and cavity modes

Box 2. Panorama of cavities and cavity modes. A common Fabry-Perot cavity (Panel A) formed by two parallel mirrors supports linear modes and maintains time-reversal symmetry. Cavities employing chiral meta-surfaces support helical modes (Panel B). A possible realization of time-reversal symmetry-breaking is offered by the use of Faraday mirrors in panel (Panel C). Ring mode cavities (Panel D) sustain running waves of a chosen circular polarization and break time reversal symmetry by means of a handedness filter realizable with a combination of a Faraday rotator and polarization optics. Advanced cavities are well suited for the exploration of the physics of spin vortices and skyrmion spin textures in exciton-polariton condensates originating from the optical spin Hall effect [37], [38]. Panels A-D from Ref.[39]. Panel E: multi-mode cavity QED enabling local light-matter coupling. The schematic displays two 87Rb Bose-Einstein condensates trapped at locations x_{1} and x_{2} on opposite sides of the cavity center [40]. Panel F: schematic of a strongly interacting *polaritonic quantum dot* formed by 150 Rydberg-dressed Rubidium in a single-mode optical resonator [41]. BS: beamsplitter; D1, D2: single-photon detectors.

##### Figure 2 - Charged exciton-polaritons

Figure 2: Charged exciton-polaritons. Panels A: Two quantum wells, labeled with the indicies σ = 1, 2 and separated by a distance d, form an electron-hole bilayer in the extremely imbalanced limit. The minority species belongs to the σ = 2 layer, while the majority species at σ = 1 forms an interacting Fermi sea. U_{q} and V_{q} are, respectively, intra- and interspecies Coulomb interactions. The bilayer is located inside a planar cavity that confines the cavity photon mode C. The (blue) shaded area represents the finite-size external laser pump spot. Panel B: The same setup in a single quantum-well geometry. Here, the majority σ = 1 and minority σ = 2 species belong to the same well. Panels C and D: the particle-hole excitation process via a photon without and with) a Fermi sea, respectively. All photon-mediated transitions are approximately vertical in a cavity. Adapted from Ref [68].

##### Figure 3 - Interior and edge polaritons in van der Waals quantum materials

Figure 3: Interior and edge polaritons in van der Waals quantum materials. Panel A: charge transfer plasmon polaritons at the interface of graphene and a-RuCl3 visualized by means of nanoinfrared methods (ω= 898 cm−1 , T = 60 K). Three types of plasmonic fringes are observed: (i) edge plasmon polaritons (dark spots at along the physical boundary of graphene crystal), (ii) interior plasmon polaritons (oscillating wave pattern emanating from the boundary of graphene on the left), and (iii) defect-launched plasmon polaritons forming circular patterns in the interior of the sample. Adapted from a study by Rizzo et al. [63]. Panels B– D: nano-IR imaging of edge plasmons on graphene nanoribbons. White dashed lines mark the boundaries of the crosscut GNR. Adapted from a study by Fei et al. [120]. Panel E: nanoinfrared image of edge plasmons in a square sample of graphene. Adapted from a study by Nikitin et al. [121]. Panels F: nanoinfrared images of edge phonon polaritons in the 25-nm-thick slab of hBN. Adapted from a study by Dai et al. [108]. Panel G: edge and interior phonon polaritons in a 40-nm-thick slab of hBN [122].

##### Figure 4 - Cavity Exciton Polaritons

Figure 4: Cavity exciton polaritons. Panel A: polaritons (pink spheres with blue halo) emerge from strong coupling between the excitonic resonance in a quantum well (transparent sheet) and the photonic mode of a GaAs/AlGaAs microcavity. THz probing (blue curve) maps out the matter component of the polaritons, while photoluminesce (PL, red arrows) leaking through a Bragg mirror reveals the photonic component. Panel B: normal-mode splitting. The heavy hole 1s exciton resonance (dashed curve) and the photonic mode (dotted curve) are replaced by the upper polariton branch and lower polariton branch (UPB and LPB, respectively; solid curves). PL (thick red arrow) originates from the radiative decay of polaritons at small in-plane momenta k||. Panel C: THz absorption probes hydrogen-like intraexcitonic transitions. While the 1s state is spectrally shifted by strong light–matter coupling, the optically dark 2p exciton is not affected by the cavity. The resulting momentum dependence of the THz transition energy allows us to map out the momentum distribution of the polaritons as they relax toward k|| = 0 (green dotted arrow). From a study by Menard et al. [132]. Panel D: schematic of the valley ´ exciton polariton phenomena. The solid (gray) curves indicate LPB and UPB. The bare cavity and the exciton dispersion are shown by the black and orange dashed curves, respectively. Pump 1 is used to excite directly the exciton reservoir, whereas pump 2 excites the lower polariton branch at specific k|| and ω. The emission is collected at smaller angles. The top inset shows the valley polarization phenomena in 2D transition metal dichalcogenide (TMDC) semiconductors caused by the broken inversion symmetry. In these materials, the K and K′ points correspond to the band edges separated in momentum space but energetically degenerate. The bottom inset is a schematic of the microcavity structure with silver and a SiO2 cavity layer embedded with prototypical TMDC materials WS2. From a study by Sun et al. [133].

##### Figure 5 - Phonon Polaritons

Figure 5: Phonon polaritons, hybrid plasmon–phonon polaritons, and image polaritons. Panel A: dispersion of phonon polaritons in ionic crystals predicted by Huang (in a study by Sun et al. [133]. In the original publication, Huang did not use the term polariton. Panel B: calculated dispersion of the hyperbolic phonon polaritons in hBN (HP2). Panel C: calculated dispersion of the hyperbolic phonon polaritons in h-BN coupled to plasmon polaritons in the graphene layer and forming hyperbolic plasmon–phonon polaritons (HP3) and surface plasmon–phonon polaritons (SP3). Adapted from a study by Bezares et al. [167]. Panel D: concept of image polaritons at the interface of hBN and a metal. From a study by Yoo et al. [168].

##### Figure 6 - Moire Polaritons

Figure 6. Moire polaritons and topological phonon polaritons in twisted van der Waals materials. Panel A: atomic force friction image of the graphene/hBN structure at the boundary between the moiré-superlattice and plain graphene (marked in Panel B). Moire reconstruction leads to a periodic pattern with the periodicity of 14 nm. Scale bar 1 mm. Panel B: nano-infrared image of the graphene/hBN structure. Darker contrast occurs in the moire region. The analysis of plasmon polariton fringes along the boundary between moire superlattice and plain graphene allows one to reconstruct the gross feature of the altered electronic structure in the moire superlattice region. Adapted from Ref. [243]. Panel C: Nano-infrared image of plasmon polaritons interference patterns in a moiré superlattice formed by twisted layers of graphene. The dashed hexagons represent the boundaries of a single unit cell. From Ref. [286]. Panel D: Plasmon polariton superposition model, which accounts for the gross features of the image in C. Panel E: Schematic of the nano-IR imaging showing an AFM tip illuminated by a focused IR beam. Panel F: nano-infrared image of moire suprlattice pattern in hBN. The contrast is formed by the shift and broadening of the phonon polariton resonance. Adapted from Ref. [287]. Panel G: nano-infrared image of phonon polaritons in a twisted structure of MoO_{3} slabs rotated by q =20^{0}, revealing complex wavefront geometry. Ref. [288]. Panel H: topological phonon polaritons in twisted MoO_{3} slabs rotated by q =77^{0}. From Ref. [289].

##### Figure 7 - Ultra-fast plasmonic effects in van der Waals materials

Ultra-fast plasmonic effects in van der Waals materials. Panel A: nano-infrared spectroscopy and imaging of switchable plasmon polaritons in black phosphorous (bP) semiconductor. Left: experimental schematics. Middle: Band structure of bP. Orange arrows indicate electron-hole pairs excited by a near-infrared pulse centered at a wavelength of 1,560 nm. The curved black arrows indicate carrier cooling towards the band extrema. Right: Ultrafast pump–probe dynamics of the scattered near-field intensity normalized to the signal at the negative delay time (equilibrium). The SiO_{2} substrate (blue points) shows no dynamics, whereas the SiO_{2}/bP/SiO_{2} heterostructure (black points) features a strong pump–probe signal. Adapted from a study by Eisele et al. [340]. Panel B: Methods for controlling plasmons in van der Waals materials and the corresponding timescales. Static and persistent tuning methods are displayed in the blue boxes; dynamical control methods are displayed in the orange ones. The yellow boxes show the dephasing times (*τ*) of plasmons and magneto-plasmons in van der Waals materials along with characteristic timescales of electron tunnelling in these systems. The green boxes represent timescales pertinent for various photonics technologies. The box with the dashed green outline indicates the desired timescales for future ultrafast plasmonic circuits. NEMS: nanoelectromechanical systems; G: graphene. Adapted from a study by Basov et al. [363].

##### Figure 8 - Tip enhanced spectroscopy of plexcitons

Figure 8: Tip-enhanced spectroscopy of plexcitons. Panel A: the strongly confined ∣Ez∣ field in a plasmonic nanogap cavity surrounding a single isolated CdSe/ZnS quantum dot (QD) and a tilted Au tip induce coupling between the plasmon and exciton. Panels B: Measured PL spectra for the QD, cavity plasmon polariton, weakly coupled system (WC) and strongly coupled states (SC) with coupling strength g = 141 meV. A Lorentzian lineshape representing the redshifted plasmon resonance in the presence of the QD is calculated from the fitted values (SPPQD) [375].

##### Figure 9 - Infrared nano-imaging of polaritonic waves

Figure 9: Infrared nanoimaging of polaritonic waves. Panel A: nano-IR image of the interference pattern of surface phonon polaritons on a SiC launched by circular Au discs [309]. Panel B: prediction of inplane negative refraction between plasmon polaritons in graphene and phonon polaritons in an hBN slab [432]. Panel C: nano-IR imaging of polariton evolution and canalization in an hBN metasurface [236]. Panel D: optical images of the laser-written metalense (bottom). Nano-IR image of revealing focusing of phonon polaritons at 1452 cm−1 [433]. Panel E: refraction of graphene plasmon polaritons at the prism formed by bilayer graphene [434].

##### Figure 10 - Exciton Polaritons

Figure 10: Panel A: schematic of MoS2/WSe2 heterobilayer nanolaser integrated in photonic crystal cavity [380]. Panel B: polaron–polaritons in TMDC semiconductors. Schematic to illustrate the conduction and valence band structure and optical selection rules of monolayer MoSe2 close to the K and K′ points. An exciton in the K valley interacts with conduction band electron–hole pairs in the Fermi sea of the K′ valley to form an intervalley polaron. From a study by Bing Tan et al. [445] Panel C: experimental setup for the exploration of propagating solitons in the system of microcavity exciton polaritons [446].

##### Figure 11 - Polariton waveguide QED

Figure 11. Polariton waveguide QED*. *Panel A: Emergence of bound atom-photon dressed states in 1D waveguides with finite bandwidth. The slow-light waveguide can be modeled as a large array of coupled optical resonators with nearest-neighbor coupling J. Lower left: band structure of the waveguide without atoms. Lower right: single-photon spectrum as a function of the atom-photon coupling g in the case of a single atom (with ωa = ωc) coupled to the waveguide, showing the emergence of bound states. Reproduced from a study by Calajo et al. [479]. Panel B: Photonic crystal for phonon-polaritons in LaTaO_{3}. Top left: optical microscope images of the photonic crystal patterns. Top right: schematic of pump-probe experiments. Bottom left: Space–time plot of THz waves generated directly inside a square photonic crystal. The edges of the image are the edges of the photonic crystal. Bottom right: Dispersion diagram obtained from a 2D Fourier transform of the space–time plot in bottom left panel. The region highlighted in yellow represents the light cone. The regions highlighted in orange show the locations of the leaky modes.Adapted from a study by Ofori-Okai et al. [480]. Panel C: tunable and switchable photonic crystal for surface plasmon polaritons in graphene. Top: Schematic of a photonic crystal comprised of a graphene monolayer fully-encapsulated by hexagonal boron nitride on top of an array of SiO_{2} pillars. Pixelated gate insulator implemented in the form of nano-pillars enables the local modulation of the carrier density and therefore of the plasmonic density of states. Bottom left: near-field nano-IR image of plasmonic standing waves for a structure in the top panel. Scale bar 400 nm. Bottom right: Calculated plasmonic band structure as a function of wave-vector k and average carrier density ns. A vertical cut parallel to the ω-k plane (back panel) generates the plasmonic band structure at fixed carrier density n_{s}=5.5×10^{12} cm^{2}. The dashed lines mark the range of a complete plasmonic bandgap. A horizontal cut parallel to ns-k plane (bottom panel) generates the plasmonic dispersion as a function of average carrier density ns and wave-vector k, at laser frequency ω= 904 cm^{−1}; a complete bandgap is evident for carrier density around n_{s}=5.5×10^{12} cm^{2}.

##### Anderson-Higgs polaritons

##### Bardasis-Schrieffer (BaSh) polaritons

*Bardasis-Schrieffer* (*BaSh) polaritons*. The matter constituent of BaSh polaritons is associated with the fluctuations of sub-dominant order parameter in superconductors[15, 16], charge density wave systems [13] and excitonic insulators[17]. This novel theoretical concept still awaits experimental confirmation. The requisite experiments include nano-spectroscopy and nano-imaging of polaritonic dispersion in the terahertz (THz) frequency range below the energy gap of superconductors. These are challenging scanning probe measurements, as they have to be carried out at cryogenic temperatures. Nano-THz imaging at cryogenic temperatures have been recently fulfilled[18] paving the wave to the exploration of polaritonic phenomena in superconductors (see also *Cooper pair plasmon polaritons* and *Josephson plasmon polaritons*).

##### Berreman polaritons

*Berreman polaritons*:* *Phonon polaritons in anisotropic materials and multilayer structures, also referred to as epsilon-near-zero or *ENZ polaritons **[25]–[27]*. ENZ materials, artificial structures and nano-cavities reveal exotic electromagnetic responses with a broad range of technological applications [27]–[31]. For example, ENZ nano-cavities facilitate ultra-strong coupling between plasmonic and phononic modes [31] as well as the so-called photonic doping [32].

##### Berry plasmon polaritons

##### Bose-Hubbard polaritons

*Bose-Hubbard* *polaritons*: cavity QED polaritons with matter component associated with transitions across the Mott gap in the system of interacting atoms [36] (see also *Mott polaritons*).

##### Bragg polaritons

*Bragg polaritons*. Bragg reflectors (Box 1 Panel G, Figs. 2 and 4) are routinely utilized to implement polaritonic cavities. Bragg polaritons pertain to systems in which multiple excitonic layers and/or quantum wells are periodically integrated in a DBR cavity [40], [42] (see also *polaritonic lattices*). The inherent anisotropy of Bragg multilayer structures may enable hyperbolic electrodynamics[43] (see *hyperbolic polaritons*).

##### Cavity polaritons

*Cavity (micro-cavity) polaritons*. Weisbuch et al. [44] devised and implemented the first semiconductor (micro)cavity device revealing Rabi splitting of exciton polaritons (Boxes 1 and 2). Semiconductor microcavites emerged as a powerful platform for the investigation of strong light-matter interaction in semiconductors [45], [46]. Microcavity structures reveal intriguing phenomena including the polariton parametric amplification [47] and its spontaneous counterpart, the parametric photoluminescence [48]. Parametric photoluminescence is a purely quantum process. An appealing attribute of polariton parametric photoluminescence is that signal-idler polariton pairs are produced in non-classical states with quantum correlations. The quest for Bose-Einstein condensation of microcavity polaritons has produced a stream of breakthrough results [49], [50] (see also e*xciton-polaritons and their condensates*). Microcavity exciton-polaritons display quantum effects including entanglement [51] and polariton blockade [52], [53] and may serve as a platform for the implementation of qubits [54].

##### Channel polaritons

*Channel polaritons* are supported by materials and structures with a straight channel cut in polaritonic medium [55]. Channel polaritons were utilized for the implementation of waveguide components including interferometers and ring resonators [56]. Polaritons guided along the nano-slit are predicted to form *hybrid polaritons* giving rise to both bonding and antibonding modes [57].

##### Charge transfer polaritons

*Charge transfer polaritons.* The formation of plasmon polaritons in graphene or semiconductors relies on the high carrier density that can be introduced by electrostatic gating [58], [59], ferroelectric polarization [60], chemical doping [61] or photo-excitation [62]. Alternatively, the requisite carrier density can be introduced by charge transfer across the interface between proximal materials with dissimilar work functions. Such charge transfer plasmon polaritons have been demonstrated for graphene residing on another van der Waals material RuCl_{3 }[63]. Experiments on metallic nano-particles show that charge transmitted between the pair of nano-particles through a conducting pathway leads to a characteristic plasmonic response [64] termed *charge transfer plasmons*. Interlayer exciton in TMDC heterostructures (e.g. MoSe_{2}/WSe_{2}) also involves charge transfer from one layer to another; the relevant microcavity polaritons [65] are classified as *charge transfer exciton polaritons*.

##### Charged polariton

*Charged polariton.* Charged polaritons posess a non-vanishing electric charge. This interesting concept was introduced in the context of the cavity exciton polaritons in GaAs/AlAs quantum wells that also hosted two-dimensional electron gas with the density n_{e}. Spectroscopic experiments in [66] have identified several distinct properties of charged exciton polaritons, including the scaling of the coupling strength analogous to the properties of atomic QED system [67]. The effective mass of charged polaritons exceeds the band mass of a GaAs quantum well by a factor of 200. Tiene et al. have theoretically demonstrated the unique utility of charged micro-cavity polaritons for exploring the physics of electron-hole systems with charge imbalance, which are difficult to access with alternative experimental methods [68]. They demonstrated how the Fermi sea of excess charges modifies both the exciton properties and the dielectric constant of the cavity medium, which in turn affects the photon component of the many-body polariton ground state (Fig.2). See also the closely related entries of *Fermi-edge exciton polaritons* and *trion polaritons*.

##### Cherenkov polaritons

*Cherenkov polaritons. *In the Cherenkov effect [69], a charged particle moving with a velocity faster than the phase velocity of light in the medium radiates light. The emitted radiation forms a cone with a half angle determined by the ratio of the two velocities. Genevet et al. demonstrated that by creating a running wave of polarization along a one-dimensional metallic nanostructure consisting of subwavelength-spaced rotated apertures that propagates faster than the surface plasmon polariton phase velocity, one can generate surface plasmon wakes that serve as a two-dimensional analogue of Cherenkov radiation [70]. The Cherenkov physics is also relevant to the properties of phonon polaritons [71], [72]. Infrared nano-imaging experiments reveal Cherenkov phonon polariton wakes emitted by superluminal one-dimensional plasmon polaritons in a silver nanowire on the surface of hexagonal boron nitride [73]. See also *Exciton-polariton X-waves* on superluminal properties in the system of exciton-polaritons

##### Cooper pairs polaritons

*Cooper pairs polaritons **(in *QMs* and cold fermionic cavity systems). Cooper pair plasmon polaritons* is an inherent attribute of superconductors. The matter component of these polaritons is associated with the superfluid density [1]. The dispersion of Cooper pairs plasmon polaritons in layered cuprate high-Tc superconductors has been investigated theoretically [12], [74] but is yet to be explored in experiments. Recently, the formalism of the Bardeen Cooper and Schrieffer theory of superconductivity has been applied to describe the quasiparticle excitations of a cold fermion system coupled to a cavity. Depending on the excitation density and atomic interaction, the excited atoms and holes and in the Fermi sea may form bound Cooper pairs strongly coupled with cavity photons. This latter kind of polaritons were also termed Cooper pair polaritons [75]

##### Dark polaritons in QM

*Dark polaritons in QMs: *polaritons characterized by a wavevector that lies beyond the light line. The lower branches of polaritons in many/most QM systems are dark by this criterion and do not couple to free space photons because of the notorious “momentum mismatch” problem (Box 1 F, H, Fig.1).Light coupling to dark polaritons can be can be mediated by nanoscale defects such as a protrusions, divots, or cracks, exploiting the high spatial frequencies inherent to these deeply subwavelength objects. Better controlled strategies can also provide the missing momentum needed for coupling to dark polaritons [76]. These include prism and grating coupling, and the use of plasmonic optical nanoantennas [77]–[84]. Notably, sharp scan-probe tips can act as such antennas [85]–[90] allowing polaritonic waves to be launched and visualized. Scanning probe antenna-based nano-optics has emerged as an indespensible research tool enabling spectroscopy and visualization of polaritons in QMs [1], [83], [91].

##### Dark-state polaritons in atomic ensembles

*Dark-state polaritons in atomic ensembles*: typically, this refers to polaritons in atomic ensembles that propagate in the regime of electromagnetically-induced transparency (EIT) [92]–[95]. The darkness arises from the photon mixing strongly with a collective atomic excitation, resulting in a state with only a minute photonic component. See also *EIT polaritons* below. In ordered atomic arrays, dark (also often reffered to as subradiant) states emerge due to interference in photon emission and absorption. At the single photon level, these darks states are collective spin excitations with a wave-vector that lies beyond the light line, preventing the coupling with radiation modes (exactly the same phenomenon of “momentum mismatch” described above for QMs) [96]–[99]. Polaritons arising in atomic lattices have applications in quantum information storage and processing [98].

##### Demons

##### Dirac plasmon polaritons

*Dirac* *plasmon polaritons* are formed by hybrids of infrared photons with Dirac electrons in graphene [58], [59], [102], [103]. Direct nano-imaging experiments uncovered extraordinarily long propagation lengths of highly confined Dirac polaritons and have established fundamental limits underlying their decoherence and losses [104].

##### Dyakonov surface polaritons

*Dyakonov surface polaritons*: the surface modes that propagate along the interface between isotropic and uniaxial materials [105]–[107]. A special case of Dyakonov polaritons is realized in anisotropic crystals of layered van der Waals materials. One example is that of the *hyperbolic surface phonon polaritons* propagating along the edges of slabs prepared from hexagonal boron nitride [108]–[110].

##### Edge magneto-plasmons

*Edge magneto-plasmons.* Two-dimensional (2D) electron gas subjected to the magnetic field normal to the plane of the 2D conductors reveals two distinct field-dependent resonances: the cyclotron resonance mode with frequency increasing with the magnetic field and another mode that redshifts with the applied field. The latter mode has been linked to the edge plasmons of the charged sheet and can be viewed as the 2D analog of surface plasmons in three dimensional (3D) systems [111]. Specifically, edge magneto-plasmons can propagate along the physical boundary of the 2D conductors [112], [113]. Edge magneto-plasmons constitute a spectacular manifestation of the dynamical Hall effect. Edge magneto-plasmons are chiral. Their chirality is a direct implication of the applied Lorentz force [114]. Graphene reveals rich plasmonic phenomena in the presence of magnetic fields [115]–[119].

##### Edge plasmon polaritons

*Edge plasmon polaritons*: one-dimensional plasmonic modes propagating along the physical boundaries of two-dimensional materials (Fig.3). Edge plasmon polaritons reveal an approximately 10 % shorter wavelength compared to the interior of the plasmonic medium [121]. Qualitatively, the shorter wavelength can be ascribed to the effective reduction of the Drude weight since free carriers exist only on one side of the physical boundary. *Dyakonov hyperbolic phonon-polaritons* are a lattice analog of edge plasmon polaritons. Berini reported on an in-depth numerical analysis of edge and corner plasmon-polariton modes in thin conducting slabs [123].

##### Whispering-galley polaritons

##### EIT in nanoplasmonic structures

##### EIT polaritons propagate

*EIT polaritons* propagate in atomic systems under conditions of electromagnetically induced transparency (EIT). A remarkable aspect of EIT polaritons is that they can be slowed down to 10-s of meters per second [134] or even brought to a standstill [135], [136]. EIT polaritons can be dark (decoupled from radiation, more “atom-like”) or bright (coupled to radiation, more “photon-like”). The darkness/brightness of the polaritons is controlled by an external laser beam. EIT polaritons can be strongly interacting, if coupled to Rydberg states (see *Rydberg polaritons* below). The EIT phenomenon is also observed in materials and nanostructures.

##### ENZ Polaritons

*ENZ polaritons*: epsilon near zero or ENZ polaritons are equivalent to *Berreman polaritons* above.

##### Exciton-polaritons and their condensates

*Exciton-polaritons and their condensates. * Exciton-polaritons are bosonic quasiparticles originating from photons hybridized with hydrogen-like bound electron-hole pairs. Semiconductor microcavities (Box 1 and Fig. 4A) offer an outstanding platform for the investigation of exciton polaritons and the attendant strong light-matter coupling. Provided a high-quality microcavity is nearly resonant with an excitonic transition, trapped photons may be emitted and reabsorbed multiple times before being lost to dissipation or cavity leakage. Absorption and re-emission of photons in the cavity gives rise to light-matter mixed eigenstates [44]. When sufficiently long-lived, exciton-polaritons may form coherent quantum states [137]–[139]. Bose Einstein condensates (BEC) of exciton polaritons are appealing quantum liquids in part because their coherent state is created and controlled by light [140]–[142]. The binding energies of excitons in organic molecules [143], transition metal dichalcogenides and lead halide perovskites can be as high as 0.75 eV [20], [144]–[151] these extraordinary high binding energies underlie the theoretical predictions of condensation and superfluidity at T=300 K [152]–[154]. BECs of exciton polaritons were predicted to form spatially and temporarily ordered states: time crystals [155]. Exciton-polariton condensates may also enable energy-efficient lasers [156].

##### Exciton-polariton X-waves

*Exciton-polariton X-waves*: wavepackets of exciton-polaritons that sustain their shape without spreading, even in the linear regime. In Ref. [157]. self-generation of an X-wave out of a Gaussian excitation spot is obtained via a weakly nonlinear asymmetric process with respect to two directions of the nonparabolic polariton dispersion. Notably, X-waves were found to propagate with superluminal peak speed with respect to the group velocity of the polaritonic system.

##### Fermi edge exciton polaritons

*Fermi edge exciton polaritons* [158], [159] are observed in microcavities where the active semiconductor is heavily doped to form the Fermi edge. Fermi edge exciton polaritons are formed of electron hole pair excitations involving electron and hole states with in-plane wave vectors around the Fermi edge: *k _{IIe}=k_{IIh}~k_{F}* where k

_{F}is the Fermi wavevector. In some litrature this latter form of polaritonic states are reffered to Mahan exciton polaritons [160] recognizing a prediction of excitonic bound states in doped semiconductors beyond the critical density of the insulator to metal transition states by G. Mahan [161], [162]. See also

*Quantum Hall polaritons*below.

##### Floquet polaritons

*Floquet* *polaritons. *The concept of Floquet engineering refers to the control of a system using a time-periodic optical field and is being broadly applied in atomic physics as well as in the field of quantum materials [163]. The notion of Floquet polaritons pertains to polaritons in a system of Floquet-engineered atomic states [164] or electronic states in solids [165]. The concept of Floquet engineering by time-period optical fields has been extended to coherent phonons in quantum materials [166]. *Chiral Floquet polaritons* are predicted [39] to form in chiral cavities in which fundamental matter symmetries are broken (Box 2).

##### Frenkel exciton polaritons

*Frenkel exciton polaritons*. The matter constituent of these polaritons originates from Frenkel excitons characterized by the Bohr radii of the same order as the size of the unit cell. Frenkel exciton polaritons are common in organic semiconductors [169]. The high exciton binding energy (~eV) and large oscillator strength may lead to room temperature exciton-polariton condensates [146], [170], [171].

##### Fuchs-Kliewer interface polaritons

*Fuchs-Kliewer* *interface polaritons*: phonon polaritons occurring at surfaces and interfaces [172] with the matter part originating from Fuchs-Kliewer surface phonons [173]. Huber et al. [174] employed nano-infrared methods to visualize propagating Fuchs-Kliewer surface phonon polaritons in SiC. Surface phonon polaritons are observed in insulating and semiconducting materials including hBN [92], [93] and SiC [175]–[178] GaAs [179], and many others.

##### Helical plasmon polaritons

*Helical plasmon polaritons*: were predicted to form in topologically nontrivial Weyl semimetals [180]. Plasmon polariton dispersion may enable the detection of a chiral anomaly: a charge imbalance between the Weyl nodes in the presence of electric and magnetic fields [181]. The Fermi surface of Weyl semimental features open disjoint segments – the Fermi arcs – associated with the topolical surface states. The resulting *Fermi arc plasmon polaritons* are predicted to be chiral and to reveal unidirectional propagation [182]. Helical plasmon terminology was also applied to describe one-dimensional plasmon polaritons associated with the helical state in domain walls of topologically nontrivial conductors including anomalous quantum Hall systems [183]. Helicity-dependence of plasmon polaritons is discussed in the context of unidirectional propagation in plasmonic meta-structures controlled by the circular polarization of light [184]–[186].

##### Hopfield polaritons

*Hopfield polaritons*: a bold theoretical concept of light-matter hybridization proposed by John Hopfield in his doctoral thesis back in 1958 (Ref. [186]). Hopfield also co-authored the first experimental paper on polaritons devoted to the study of phonon polariton dispersion in GaP by means of Raman scattering [187]. Other early contributions to the theory of polaritons (short of introducing this term) were made by Fano [188], Huang [189] and Tolpygo [190].

##### Hybrid polaritons

*Hybrid polaritons.* Different types of polaritons hosted by the same material are prone to hybridization [191]. For example, *intersubband polaritons* and *phonon polaritons* hybridize in semiconductor quantum wells [192]–[194]. Hybridization can also occur in multi-layered structures. In all-dielectric layered structures phonon polaritons associated with the neighboring layers couple to form hybrid modes [82], [195], [196] Semiconductor heterostructures [197], [198] and especially Van der Waals heterostructures offer a fertile platform for the implementation of hybrid polaritons [199], [200]. One such example (Fig.5 B,C) is graphene surrounded by insulating layers of hexagonal boron nitride hBN or silicon dioxide. Plasmons associated with graphene layers hybridize with phonon polaritons in proximal SiO_{2} or hBN layers to form *plasmon-phonon polaritons **[58], [59], [167], [201]*. Hybrid polaritons at the interface of graphene with high-T_{c} superconductors were proposed as a tool to probe Anderson-Higgs electrodynamics *[202]*. Hybrid polariton at the interface of graphene with a charge density wave materials were theoretically proposed to “melt” the density wave order [203]. Hybrid modes produced by plasmons in graphene and molecular vibrations of absorbates on the graphene surface may enable high-selectivity sensing mechanisms [204], [205]. A special case of hybrid modes is *hybrid longitudinal-transverse phonon polaritons **[206]*. Polaritonic heterostructures with phase change materials enable persistent switching of polaritonic response under thermal and optical stimuli *[207]*.

##### Hyperbolic polaritons

*Hyperbolic polaritons. *Anisotropic media are predicted to support an interesting class of polaritonic light-matter modes referred to as “hyperbolic” because their iso-frequency surface is a hyperboloid [201], [208]–[215]. These modes exist over a range of frequencies where the in-plane permittivity and the out-of-plane (c-axis) permittivity are of the opposite sign. Hyperbolic electrodynamics and hyperbolic polaritons can originate from a variety of physical processes including phonons [207], [211], [216]–[225] inter-subband transitions in quantum wells [226]–[228] plasmons [208], [214], [229]–[232] excitons [233] and Cooper pairs (see *Cooper pair polaritons*). Hyperbolic polaritons dramatically enhance the local photonic density of states and are predicted to give rise to strong nonlinearities [234]. Hyperbolic polaritons enable canalization imaging [235] with image effectively transferred by high-momentum sub-diffractional polaritonic rays from back to front surface of the polaritonic medium [236]–[239].

##### Image Polaritons

*Image polaritons:* virtual polariton modes produced by image charges at the interface of a polaritonic medium and a metal. Lee et al. have experimentally demonstrated low loss response of image polaritons at the interface of hBN separated with a thin spacer from a metallic substrate [240] (Fig.5D).

##### Interband Polaritons

*Interband polaritons. *The matter constituent of these polaritons originates from contributions of the optical response associated with transitions across the energy gap in the electronic spectrum of a material. These include transitions across the energy gap in semiconductors [241] and superconductors or transitions involving mini-bands/flat-bands in moire superlattices of van der Waals materials [242]–[245] (see also *Moire polaritons*). The frequency dependence of s_{2}(w)xw, where s_{2}(w) is the imaginary part of the complex conductivity, is informative for the analysis of interband polaritons [243], [246]. Spectra of s_{2}(w)xw reveal a series of steps separated by plateaus, with each step uncovering the energy scale associated with separate interband contributions. In the limit of wà 0, the product s_{2}(w)xw quantifies the spectral weight of intraband processes to the plasmon polaritons. Interband effects play a central role in theoretical proposals for the implementation of population inversion [247], gain and *superluminal plasmon polaritons* *[248]*.

##### Intersubband Polaritons

*Intersubband polaritons*. Dini et al. reported the first experimental observation of the vacuum-field Rabi splitting of an intersubband transition inside a planar microcavity hosting two-dimensional electron gas [249], [250]. Nonlinearities associated with intesubband transitions in semiconductors can be dramatically enhanced by in hybrid structure with plasmonic metasurfaces [251](see also *hybrid polaritons*).

##### Josephson Plasmon Polariton

*Josephson* *plasmon polariton*: an inherent attribute of strongly anisotropic layered superconductors. The matter constituent of Josephson plasmon polaritons originates from interlayer Josephson plasmon in layered superconducting materials such as cuprates [74], [252] [Stinson-JPR}. Josephson plasmons are the electromagnetic signature of three-dimensional superconductivity in highly anisotropic layered high-Tc superconductors [253]. Josephson plasma waves can be parametrically amplified under illumination with pulsed THz fields [254] paving the way for active Josephson polaritonics.

##### Kane Polaritons

*Kane polaritons*: surface plasmon polaritons formed with Kane quasiparticles. Kane polaritons were recently observed in pump-probe experiments on narrow gap II-VI semiconductors [255].

##### Landau Polaritons

*Landau polaritons*. The matter component of Landau polaritons originates from cyclotron resonances and transitions between quantized Landau levels relevant in low-dimensional electron gases subjected to high magnetic fields [256], [257]. See also *magneto-plasmon polariton*.

##### Luttinger Liquid Polaritons

*Luttinger liquid polaritons*: plasmon polaritons in one-dimensional conductors recently revealed by infrared nano-imaging of single- and multiple-wall carbon nanotubes [258]. Interacting electrons confined in one dimension are generally described by the Luttinger liquid formalism [259], [260]. Anomalous dependence of the plasmonic quality factor on gate voltage was interpreted in terms of plasmon-plasmon interaction in carbon nano-tubes [261].

##### Magnon Polaritons

*Magnon polaritons.* The matter constituent of these polaritons originate from antiferromagnetic [262], [263] and ferromagnetic resonances. In weak magnetic fields *surface magnon polaritons* are predicted to acquire non-reciprocal properties. Macedo and Camley analyzed the propagation of surface magnon polaritons in anisotropic antiferromagnets [264]. Sloan et al. predicted that surface magnon polaritons will strongly enhance the spin relaxation of quantum emitters in the proximity of antiferromagnetic materials such as MnF_{2} or FeF_{2} (Ref. [265]). Kruk et al. developed artificial structures with hyperbolic magnetic response with principal components of the magnetic permeability tensor having the opposite signs [266]. Magnetic materials also support *hybrid polaritons*, including hybrid *magnon-phonon polaritons* recently observed in ErFeO_{3}/LiNbO_{3} multilayers [267].

##### Magneto-Plasmon Polaritons

*Magneto-plasmon polaritons:* coupled modes of magneto-plasmons and THz/infrared photons [268], [269]. Theoretically predicted unconventional properties of magneto-polaritons in Weyl semi-metals include hyperbolic dispersion and photonic stop bands [270]. The nano-infrared imaging and visualization of magneto-plasmon polaritons remains an unresolved experimental challenge. Once technical obstacles are circumvented, it may become possible to directly explore both the focusing and the non-reciprocity predicted for magneto-plasmon polaritons [271]. Plasmonic system driven by intense a.c. field is predicted to reveal spontaneous summetry breaking and non-linear magnetism [272].

##### Microcavity Polaritons

*Microcavity polaritons**: *see* cavity polaritons.*

##### Moire Polaritons

*Moire polaritons. *Atomic layers comprising van der Waals materials can be re-assembled into heterostructures with nearly perfect interfaces [273]–[275]. A unique control knob specific to vdW systems is the twist angle q between the adjacent layers. Varying q forms moiré superlattices that can radically modify the electronic structure and attendant properties [276]–[280]. Plasmons, phonons and excitons are all altered in moire superlattices prompting changes of the corresponding polaritons. G/hBN[cross-Ni-Moire]. Infrared nano-imaging data display rich real space patterns of polaritonis with selected examples of moire polaritons displayed in Fig.6. Morie design principle can be applied to epitaxially grown thin films on dielectric substrates [281]. Recent experiments on interlayer excitons in TMDC heterobilayers have revealed the trapping of these excitons on the moire potential landscape [282]–[285]. When placed in an optical cavity, such moire trapped excitons may form an exciton polariton lattice and serve as analog quantum simulators (see p*olaritonic lattices and quantum simulators*).

##### Molecular Polaritons

*Molecular polaritons.* Organic semiconductors and molecules embedded in optical (nano)cavities under strong and ultrastrong coupling promote the dynamical formation of molecular polaritons: hybrid energy eigenstates composed of entangled photonic, electronic, and vibrational degrees of freedom [30], [290], [291]. Molecular polaritons were demonstrated to enhance energy transfer [292] and DC conductivity [293]. Progress with nano-structures enabled a demonstration of the strong-light matter coupling with a single molecule embedded in a plasmonic cavity [294]. Molecular polaritons enable control of optical nonlinarities via manipulations of cavity characteristics [295]. Molecular polaritons can form *hybrid polaritons* by coupling to surface plasmons [296], for example. We remark that molecular polaritons are commonly reffered to as *vibrational polaritons*.

##### Mott Polaritons(QED)

##### Mott Polaritons (QM)

*Mott polaritons (QM)* were also introduced in context of the resonant coupling between strongly correlated electrons in solid Mott insulators integrated in a single-mode cavity [299].

##### Phonon Polaritons

*Phonon polaritons:* is a collective excitation comprised of (infrared) light coupled with a polar lattice vibration. Like other polaritons, phonon polaritons can be understood in terms of an anti-crossing of the dispersion curves of light and matter constituents (Fig. 5). Early observations of phonon polaritons (see *Hopfield polaritons*) in bulk crystals and films were made using a variety of spectroscopic methods [300], [301]. More recent work [302] has focused on the generation, detection, and on picosecond polaritons dynamics [303]–[307]. By matching the phonon-polariton velocity in LiNbO_{3 }crystal to the group velocity of the fs pump pulse Yeh et al. have been able to generate intense THz fields of the order of 10 mJ energy [308]. Advanced nano-imaging/spectroscopy methods [309]–[311].were employed for the real-space visualization of phonon polariton standing ways. Phonon-polaritons play a major role in nano-scale thermal transport at nano- and meso-scales [81], [312]–[317]. Phonon polaritons in the anisotropic oxide material MoO_{3} reveal both elliptical and hyperbolic dispersions [318], [319]. The dispersion and propagation of phonon-polaritons can be controlled by nano-structuring [320] and twist-angle (moire) engineering (Fig.6). The recent discovery of parametric phonon amplification in SiC paves the way for the exploration of non-linear and active phonon-polariton phenomena [321]. *Surface phonon polaritons* (see also *Fuchs-Kliewer* interface polaritons), reveal a dispersion branch located between longitudinal and transverse vibrational modes (see *hybrid polaritons*). Dai et al. detected surface phonon polaritons in monolayers of hBN [322].

##### Plasmon Polaritons

*Plasmon polaritons*: probably the most thoroughly studied class of polaritons. A surface plasmon polariton is a transverse magnetic (TM)-polarized optical surface wave that, for example, propagates along a flat metal-dielectric interface, typically at visible or infrared wavelengths [323]–[325]. Plasmon polaritons have rich implications for technology [326]–[329]. Nonlinear [330]–[332] and quantum [333]–[337] properties of plasmonic structures are in the vanguard of current research. Plasmon-polaritons can be controlled at femto-second time scales [62], [255], [338]–[341] enabling access to novel physics and applications [342], [343]. Plasmonic waveguides have been incorporated with light-emitting materials, paving the way for integrated plasmonic and photonic structures [344]. Plasmon-polaritons have been harnessed to implement high-quality factors such as whispering gallery microcavities [345]. In parallel, many research groups are searching for new plasmonic media with the properties optimized for different classes of plasmonic effects [346]–[349]. Van der Waals materials, and especially graphene, are emerging as outstanding plasmonic media in light of their inherent tunability with different stimuli (see *Dirac plasmons*). *Acoustic plasmon polaritons* are a special example of *hybrid polaritons* whose frequency-momentum *w(q)* dispersion is predicted to be linear [350]–[354]. Acoustic plasmon plaritons have been demonstrated [355]–[357] in structures where graphene resides in close proximity to metallic surfaces. *Spoof* *surface plasmons polaritons* were introduced describe plasmon polaritons on the surface of artificial metallic structures and metamaterials [358]. *Airy surface plasmon polaritons* are the surface counterparts of non-diffracting airy waves [359] and have been demonstrated by direct nano-imaging [360]. *Chiral plasmon polaritons **[361]* were predicted to occur in twisted bilayer graphene [362].

##### Plexcitons

*Plexcitons* are a specific example of hybrid polaritons. The matter constituent of plexcitons originates from plasmon-exciton coupled modes [364]–[370]. Historically, plexciton studies have focused primarily on localized states [365], [371]. Propagating plexciton states also exist and offer potential for compact quantum information carriers a well as opportunities for mediating emitter-emitter coupling [372]–[374]. Composite structures and multilayers can feature plexcitons. An interesting recent example of plexciton study has been conducted in the setting of scanning probe nano-optical imaging and spectroscopy (Fig.8). This work by May et al [375] along with a study by Groß et al. [376] implemented the scanning optical cavities formed between a nano-optical antenna and the substrate. The authors investigated CdSe/ZnS quantum dots using this scanning cavity approach and observed plexitonic Rabi splitting of 163 meV.

##### Polaritons Parametric Amplification

*Polaritons parametric amplification, gain* *and lasing* have been demonstrated for exciton polaritons in microcavities [47], [377]–[379]. Resonant coupling between photons and excitons in microcavities can efficiently generate significant single-pass optical gains. Polaritonic lasing has been implemented and analyzed in different material systems hosting *plasmon polartions* and *exciton polaritons **[380]*. Amplification of demons [101] has been predicted as well but is yet to be experimentally demonstrated.

##### Polaritonic Chemistry

##### Polaritonic Circuits

*Polaritonic circuits, devices, arrays and systems. *Both light and matter constituents of polaritons are amenable to controls with external stimuli [386]. The use of *exciton polaritons* as building blocks for future information processing such as spin-switches [387],spin-memory [388], transistors [389], logic gates [390], resonant tunneling diodes [391], routers [392], lasers [393] has recently been demonstrated. The first polaritonic systems are also emerging and include quantum simulators and networks for neuromorphic computers [394]. Transition metal dichalcogenide material WSe_{2}, integrated into microcavity devices acts as efficient light emitting device [395].

##### Polaritonic Lattices and Quantum Simulators

*Polaritonic lattices and quantum simulators*. A variety of experimental approaches have been utilized to implement one- and two-dimensional arrays of interacting polaritons. In the field of *microcavity exciton polaritons* gate arrays, spatially dependent optical potential as well as surface acoustic waves [396], have been utilized to generate arrays/lattices [397]. One-dimensional *exciton polariton* superlattices reveal weak lasing assigned to a novel type of a phase transition in this interacting system [398]. Arrays of evanescently-coupled cavities hosting neutral atoms [399] have been proposed as quantum simulators (QS), where the photon blockade provided by the atom limits the occupancy of each cavity to one, allowing for the implementation of the Bose-Hubbard model. QSs require controllable quantum systems that efficiently simulate a Hamiltonian of interest, which may encode phases with a significant degree of entanglement and is not amenable to calculations by classical computer [400]–[405]. Lattices of exciton polaritons [400], [406]–[410] have emerged as a promising platform for QS, along with ultracold atoms [403], [411], trapped ions [412]–[414], and superconducting circuits [415], [416]. Moire superlattices of plasmon polaritons (Figure 6) present yet another example of polaritonic lattices. Moire superlattices were realized in graphene devices with nanostructured gate electrodes [417] as well as in moire superlattices of twisted graphene layers [286].

##### Polaritonic interference refraction, collimation, front shaping and waveguiding

Polaritonic interference, refraction, collimation, front shaping, and waveguiding. All these common wave phenomena are relevant to polaritons (Figure 9). In van der Waals materials, domain wall boundaries can act a polaritonc reflectors [418]–[420], or conductors [421]. Zia and Brongersma [422] demonstrated Young’s double-slit experiments with surface plasmon polaritonss. Beyond analogs of geometrical optics effects, polaritons offer at least two novel routes for image formation. First, hyperbolic polaritons enable canalization imaging [235], with images effectively transferred by high-momentum subdiffractional polaritonic rays from the back to the front surface of the polaritonic medium [236], [238], [239], [289] (Figure 9C). Second, polaritons are amenable to guiding and steering using methods of transformation optics. Polaritonic waveguides have been implemented over a broad range of frequencies from THz [423] and infrared regions to visible light. Peier et al. observed phonon–polariton tunneling across the airgap [424]. Advanced polaritonic launchers and metalenses (Figure 9D) are well suited for defining the trajectories of polaritonic surface “beams” [425], [426]. In highly nonlinear regime polaritons are predicted to display self-focusing effects and to form solitons [427].

##### Polariton-polariton interactions

*Polariton-polariton interactions.* The interaction of polaritons stems from their underlying matter constituents. In close analogy with other interacting systems, polariton-polariton interactions renormalize the dispersion and also prompt a blue shift of the emission energy as the polariton density increases [428], [429]. Polariton-polariton interaction effects have been recently demonstrated for *microcavity exciton polaritons **[430], [431]*. See also *quantum Hall polaritons**.*

##### Polaron Polaritons

*Polaron-polaritons.* In TMDC monolayers, the itinerant electrons dynamically screen exciton to form new quasiparticle branches — the attractive and repulsive polaron — each with a renormalized mass and energy [435], [436]. *Microcavity polaritons* with the matter constituent linked to these polaron branches are referred to as polaron-polaritons [435].

##### Quantum Hall Polaritons

*Quantum Hall polaritons* are a product of coupling cavity photons to the cyclotron resonance excitations of electron liquids in high-mobility semiconductor quantum wells or graphene sheets [437], [438]. The edge channels of the quantum Hall effect offer a platform for probing interference and entanglement effects in the setting of a condensed matter system since the edge states propagation is ballistic, one-dimensional, and chiral. This platform enables experimental implementation of electron quantum optics [439]–[442] and may be suitable for the realization of flying qubits. In a parallel development, Smolka et al. [443] investigated cavity *exciton polaritons* in the presence of high-mobility 2D electron gas subjected to external magnetic field and discovered novel correlated electron phases Knuppel et al. [444] reported on strong *polariton polariton interactions* in the fractional quantum Hall regime.

##### Rydberg Polaritons (QED)

# Rydberg polaritons (QED): photons dressed by highly excited atomic Rydberg states under conditions of electromagnetic induced transparency. These polaritons can either reside in a cavity or propagate throughout an atomic ensemble. In a cavity, Rydberg dressing bestows an atomic ensemble with the character of a two-level system: the excitation of a single Rydberg polariton prevents the creation of a second one, in the so-called “Rydberg blockade” regime. Under conditions of electromagnetic induced transparency, polaritons can propagate within an optically-dense atomic cloud. These polaritons can then be made to interact with each other via Rydberg dressing: the first Rydberg polariton alters the transparency condition for the second one, preventing its propagation within a certain “blockade radius” [447]–[450]. Rydberg polaritons are appealing for quantum logic functionalities [451] and for realizing synthetic materials via many-body states of light [138], [164].

##### Rydberg Polaritons (QM)

Rydberg polaritons (QM): a special example of exciton polaritons with matter constituent associated with strongly interacting Rydberg states of excitons [131]. Candidate systems include TMDC monolayers [452] and cuprous oxide, where Rydberg states with principal quantum numbers of up to n = 25 are feasible [453].

##### Soliton Polaritons

*Soliton polaritons*. Propagating wavepackets in semiconductor micorcavities are referred to as soliton polaritons (Fig.10 C). In quantum optics, topological soliton polaritons refer to composite objects made of fermions trapped in an optical soliton. The prototypical one-dimensional (1D) model of solitons posessing nontrivial topology is the model of Su-Schrieffer-Heeger (SSH) chains [454]. Variants of the SSH Hamiltonian have been emulated in the 1D lattices of *microcavity exciton polaritons **[455]* and also in the system of quantum emitters coupled to a photonic waveguide [456]. Topological phases of polritons in cavity waveguides were analysed in Ref. [457].

##### Spin Polaritons

*Spin polaritons*: this term was coined in the context of polariton microcavity diode lasers operating via injection of spin polarized currents [458].

##### Spin Plasmon Polaritons

*Spin plasmon polaritons* are relevant to the plasmonic response of spin-polarized electron gas [459]. Alternatively, spin-orbit interaction may lift the degeneracy between the spin states and give rise to transitions responsible for peculiar dispersion features of spin plasmon polaritons [460]. The surface plasmon of a helical electron liquid is predicted to carry spin and is also referred to as a spin plasmon polariton [461].

##### Transformation Optics with Polaritons

*Transformation optics with polaritons. *Transformation optics refers to a general principle for designing a complex electromagnetic medium with tailored properties by carefully crafting the spatial patterns of the local optical index [462], [463]. This general principle has been extended to polaritons [464] and polartonic cavities [465], and specifically to *plasmon polaritons* in graphene [466]. Losses present the most significant experimental roadblock for practical transformational polaritonics. Recent advances with highly confined but low-loss plasmon polaritons [104] and phonon-polaritons [211] fulfill important experimental preconditions for the realization of transformation optics ideas in polaritonic systems.

##### Tamm Surface Plasmon Polaritons

*Tamm surface plasmon polaritons* are associated with Tamm states at metallic surfaces [467]. Common surface *plasmon polaritons* are formed with a TM (transverse magnetic) polarization at the boundary of metallic and dielectric surfaces and lie to the right of the light cone. Tamm polaritons are found with both TM and TE (transverse electric) polarizations and their dispersion can be within the light cone [468], [469].

##### Trion Polaritons

##### Tunneling Plasmon Polaritons

*Tunneling plasmon polaritons* were predicted [473] and observed [474] in an atomically thick tunable quantum tunnelling devices consisting of two layers of graphene separated by 1 nm of h-BN. By applying a bias voltage between the graphene layers one creates an electron gas coupled to a hole gas. Even though the total charge of the devices is zero, this system supports propagating graphene plasmons.

##### Valley Polaritons

*Valley polaritons*. The matter constituent of these polaritons originates from valley polarized excitons in TMDC semiconductors (Fig.4B and *exciton polaritons*). The electronic structure of two-dimensional TMDC semiconductors endows this class of materials with the spin–valley degree of freedom that provides an optically accessible route for the control and manipulation of electron spin [475]–[477].

##### Vibrational polaritons

*Vibrational polaritons: *see *molecular polaritons*.

##### Wannier or Wannier-Mott Polaritons

*Wannier* or *Wannier-Mott* *polaritons* borrow their matter part from Wannier excitons in semiconductors [478].

##### Waveguides and photonic crystals for polaritons

*Waveguides and photonic crystals for polaritons. *Waveguides and photonic crystals allow one to design and control the properties of photons, and thus of polaritons, both in quantum optics and quantum materials. In waveguide QED, different type of emitters (neutral atoms, quantum dots, color centers, superconducting qubits) are coupled to a one-dimensional (1D) optical channels [481], such as fibers [482], [483], photonic crystals [484], [485], and transmission lines [486], [487] (Box 1 and Fig.11 A). Channel with a band gap give rise to atom-photon bound states (i.e., polaritonic bound states) provided the atomic resonance frequency is close to the band edge. Beyond the band-edge, photons are bound to the atoms, forming localized polaritonic cavities that can be harnessed for realizing quantum simulation and quantum information processing (Box 1). If the coupling between photons and atoms is strong enough, bound states emerge even if the atomic resonance frequency lies inside the band (i.e., as a “bound states in the continuum”) due to multiple scattering [479], [488]. In the field of quantum materials, photonic-crystals structures were fabricated using common phonon-polariton oxide systems LiTaO_{3} and LiNbO_{3} (Fig. 11B). Pump-probe experiments in Fig. 11B revealed the key attributes of the dispersion control by these periodic structures. A significant deficiency of conventional photonic crystals is that they do not allow for dynamical dispersion engineering. Xiong et al. circumvented this limitation and demonstrated a broadly tunable two-dimensional photonic crystal for *surface plasmon polaritons *[cross-ref-xiond]. Infrared nano-imaging revealed the formation of a photonic bandgap and an artificial domain wall which supports highly confined one-dimensional plasmonic modes.

##### Zenneck-Sommerfeld waves and Norton waves

*Zenneck-Sommerfeld waves and Norton waves*: an early example of a guided electromagnetic wave at the interface of media with negative and positive dielectric function Refs. [489]–[491] the same condition that is required for the formation of polaritonic modes in THz, infrared and optical frequencies. The original prediction of Zenneck-Sommerfeld waves pertained to the radiofrequency wave at the interface of air and the earth. In this analysis the surface of the Earth was regarded as a lossy dielectric. The concept of Zenneck-Sommerfeld waves and closely related Norton waves has been applied to a broad class of wave patterns on the surface of metallic [492]–[495] and dielectric materials [496].

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