Topological Polaritons

The interaction between light and matter can give rise to novel topological states. This principle was recently exemplified in Floquet topological insulators, where \emph{classical} light was used to induce a topological electronic band structure. Here, in contrast, we show that mixing \emph{single} photons with excitons can result in new topological polaritonic states --- or"topolaritons". Taken separately, the underlying photons and excitons are topologically trivial. Combined appropriately, however, they give rise to non-trivial polaritonic bands with chiral edge modes allowing for unidirectional polariton propagation. The main ingredient in our construction is an exciton-photon coupling with a phase that winds in momentum space. We demonstrate how this winding emerges from spin-orbit coupling in the electronic system and an applied Zeeman field. We discuss the requirements for obtaining a sizable topological gap in the polariton spectrum, and propose practical ways to realize topolaritons in semiconductor quantum wells and monolayer transition metal dichalcogenides.


Introduction
The idea of creating topological photonic states was established in 2008 [1,2]. In a seminal work, Haldane and Raghu proposed to generate the analog of quantum-Hall states in photonic crystals with broken time-reversal symmetry. Shortly thereafter, this concept was experimentally demonstrated for electromagnetic waves in the microwave domain [3,4]. Topological photonic states at optical frequencies, however, suffer from the fact that the magnetic permeability is essentially µ = 1 in this regime [5], which renders the magneto-optical response allowing to break time-reversal symmetry very weak. Proposed alternatives include the use of time-periodic systems [6,7], coupled optical resonators or cavities [8][9][10][11][12][13][14], as well as metamaterials [15][16][17]. Despite this progress, the realization of topological photonic states at optical frequencies remains challenging.
Here we pursue a new direction motivated by the concept of Floquet topological insulators [18], which rely on the specific mixing of two topologically trivial fermionic bands made possible by the absorption/emission of photons. In contrast to using photons to generate a nontrivial topology, we ask whether one can reverse this concept and create a non-trivial photon topology with the help of electronic degrees of freedom. We show that this is indeed possible by coupling photons to semiconductor excitons. Although both of these ingredients are ordinary (non-topological) by themselves, their combination can lead to new (quantum-Hall-like) topological states of polaritons which we call, in short, "topolaritons". We consider a setup where a finite spin-orbit coupling mixes photons and excitons in a non-trivial way. In combination with time-reversal symmetry breaking for the underlying semiconductor (e.g., by a Zeeman field), we then demonstrate that it is possible to generate a non-trivial topology.
In contrast to quantum Hall states of fermions, the topology of the bosonic (polariton) system that we consider is not a ground-state property. Instead, it is charac- Figure 1: Schematic view of a system with topological polaritonic bands. A chiral polaritonic mode consisting of a mixture of bound particle-hole pairs (excitons) and photons is found at the edge of the system. terized by topological bands in the excitation spectrum. The main signature of this non-trivial topology is a bulk gap in the excitation spectrum with chiral edge modes as the only in-gap states (see Fig. 1). These edge modes provide a new realization of a controllable one-way waveguide for photons [3,4,[19][20][21][22][23]. More conceptually, our proposal allows to realize topological photons at optical frequencies and, to the best of our knowledge, constitutes the first example of a topological hybrid state treating light and matter degrees of freedom on the same footing.

Topolaritons
Polaritons are superpositions of photons and excitons which can be described by a Hamiltonian of the form where the operatorsâ † q andb † q create photons and excitons with momentum q, respectively. We assume that the excitons and photons are both confined to two dimensions, i.e., q = (q x , q y ). For the excitons, this can be achieved using a quantum well, while photons can be trapped using a microcavity or waveguide. The exciton q � q P± C �� ������� C �� ������ dispersion ω X q = q 2 /2m X + ω X 0 (setting = 1) describes its center-of-mass motion, while the energy gap ω X 0 for creating an exciton is given by the difference between the bare particle-hole excitation gap and the exciton binding energy. We denote the dispersion of the cavity photon by ω C q . The crucial ingredient for generating topolaritons is the exciton-photon coupling g q which describes the creation of an exciton by photon absorption and vice versa. Here we require g q to wind according to where g q is the amplitude of the exciton-photon interaction (or Rabi frequency), m is a non-zero integer, and θ q denotes the polar angle of q.
To reveal the non-trivial topology, we diagonalize the Hamiltonian (1) in terms of polariton operatorŝ , where the ± sign refers to the upper and lower polariton band, respectively (see Fig. 2). The vectors e P ± q describe the relative strength between the photon and exciton components of the polariton wavefunction and can be interpreted as a spinor. The non-trivial form of the coupling (2) leads to a winding of this spinor of the form where spinor of the lower polariton band "flips" from photonic to excitonic (vice versa for the upper polariton) far away from the resonance as described by the limits β q=0 = −1 and β q→∞ = 1. Combined with the winding e imθq , this flip leads to a full wrapping of the unit sphere by the spinor e P ± q , thereby leading to a non-trivial topology in full analogy to fermionic topological systems. This can be confirmed by calculating the Chern number of the upper and lower polariton bands. Indeed, we find C ± = ±m.
A consequence of the non-trivial Chern number is the presence of chiral polaritonic edge modes. In our setting, edges are defined by the confinement of the excitons and photons. While excitons are naturally restricted to the quantum well, photons can be confined, e.g., by using reflecting mirrors or a suitable photonic bandgap at the edges of the system. Another possibility would be to confine the photons in a dielectric slab waveguide ending at the system edges.
Besides the existence of bands with non-trivial Chern numbers, the stability of the edge modes also requires a global energy gap (i.e., present for all momenta) between the upper and lower polariton bands. Due to the winding behavior of g q [see Eq. (2)], any lattice model will have a vanishing g q=0 to avoid any singularity (see also Methods). In that case the minimum of the upper polariton branch is pinned to the bare exciton at q = 0 while the lower polariton branch approaches the bare exciton dispersion for large q. For a positive exciton mass, it thus appears impossible to open a gap. Although a small gap can in principle be opened by considering a negative exciton mass m X < 0, as depicted in Fig. 3, we present below a more realistic and efficient way to realize topolaritons. We first discuss how to obtain a winding exciton-photon coupling, and show that a sizable topological gap can be opened in the presence of a periodic exciton potential.

Realizing topolaritons
Winding coupling. Promising candidates for the realization of topolaritons are semiconductor quantum wells embedded in photonic waveguides or microcavities (Fig. 4). One of the main requirements is the presence of a single two-dimensional exciton mode and finite spinorbit coupling. Here we start from a quantum well described by a single block of the BHZ-like Hamiltonian [24] in the topologically trivial regime, i.e., where M > 0 describes a trivial bandgap, σ i are Pauli matrices corresponding to states with different total angular momentum (1/2 and 3/2 for HgTe-based quantum wells [24]), and A is the corresponding spin-orbit coupling strength. The standard BHZ Hamiltonian consists of two blocks of the form H QW related by time-reversal symmetry. Here we break this symmetry and isolate a single of these blocks by applying an external magnetic field in the z-direction. A finite Zeeman field can shift the exciton energy when the conduction and valence bands have . Two edge modes are found within the gap, each localized at one of the system's boundaries. The color scale encodes the edge localization, a more intense red (blue) color indicating a larger weight of the wavefunction at the edge at y = 0 (y = L). For the construction of the lattice model, we choose quadratic dispersions q 2 /2mX +ω X 0 and q 2 /2mC for the exciton and photon, respectively, replacing q 2 x,y → 2 − 2 cos(qx,y) (note that a quadratic photon dispersion with mC = qres/2c correctly reproduces the original linear dispersion close to resonance at qres). Similarly, we model the winding coupling by gq = g[sin(qx) + i sin(qy)]. The system size is 40×40 (with lattice constant a = 1) and the other relevant parameters are given by mX = −500, mC = 1, ω X 0 = 0.25, and g = 0.01. different g-factors (see, e.g., [25] for HgTe-based quantum wells). Below we assume that the resulting energy splitting between excitons belonging to different blocks is large enough so that one can focus on a single one of them. We note that this strict separation is only justified provided that the Zeeman splitting is larger than the exciton-photon coupling. Our results, however, remain valid for much weaker Zeeman fields. Taking into account both excitons, we have verified that the Zeeman splitting must only exceed the topological gap in order to obtain a non-trivial topology. For reasonably large gfactors (> 10), this can be achieved with magnetic field weaker than 1T. We remark that orbital effects of the magnetic field can be neglected as long as the magnetic length is larger than the size of the excitons, as we shall assume in what follows.
When adding electron-electron interactions to the quantum-well Hamiltonian (5), excitons form as bound states of conduction-band particles and valence-band holes. The Bohr radius a 0 of such "particle-hole atoms" is typically of the order of 1 − 10nm, with binding energies of the order of 10 − 100meV (see, e.g., [26]). Expressed in terms of the creation operatorsĉ † c,k andĉ v,k for the conduction-and valence-band eigenstates, the corre-sponding exciton operator takes the form (6) and obeys bosonic commutation relations for low exciton densities (note that we assume equal masses for particles and holes, for simplicity). The exciton wavefunctions φ n (k) are (the Fourier transform of) hydrogen-atom-like wavefunctions in the relative coordinates of the bound particle-hole pairs.
For the photonic part of the Hamiltonian, we assume that the photons are confined to two dimensions by a waveguide or a microcavity, with the semiconductor quantum well being either in direct proximity or inside the waveguide/microcavity. The two-dimensional photonic modes are either of transverse electric (TE) or transverse magnetic (TM) nature. To open a gap in the polariton spectrum, we want to target a regime where a single photon mode couples to a single exciton mode. This can be realized by considering the limit of large in-plane momenta in which the electric field of the TM mode points predominantly out of the plane. Since two-dimensional excitons only couple to in-plane electric fields, we can then neglect the coupling to the TM modes. In practice, the coupling of excitons to TM modes will lead to a mixing of the resulting edge states with the TM modes. The edge states, however, will remain essentially unchanged as long as the TM coupling is much smaller than the TE coupling. The presence of TM modes within the gap can in principle be avoided using a photonic crystal with a tailored TM bandgap.
Although polaritons have been observed in photonic crystals [27], we focus on the large in-plane momentum limit to simplify the discussion. The quantized vector potential describing the TE modes then takes the form whereâ † q creates a photon of momentum q, e q⊥ = (−q y , q x )/q describes the direction of the in-plane electric field perpendicular to the propagation direction q, and F q = e 2 / 2 wω C q 1/2 where e denotes the electron charge, the dielectric constant, and w the width of the waveguide/microcavity in the z-direction. The coupling of the photons to the quantum well can be incorporated by the standard substitution k → k + eÂ in the Hamiltonian (5). The spin-orbit coupling in the quantum well combined with the locking of the photon electric-field direction (e q⊥ ) to q then leads to a winding excitonphoton coupling. For s-wave excitons and provided that A/(M a 0 ) < 1 (see Methods), we find an exciton-photon Hamiltonian of the form (1), with a winding coupling terms that are cubic or of higher order in momentum (see Eq. (5)). A higher winding number |m| = 2 can also be achieved starting from polaritons that propagate mostly in the z-direction, coupled to circularly polarized light. We discuss this alternative scheme in more detail in the Supplementary Information. As far as higher winding numbers are concerned, it would also be interesting to extend our proposal to systems in which excitons already have a non-trivial topological nature [28][29][30].
Finite topological gap. The remaining ingredient for chiral polaritonic edge modes is a finite topological gap.
One way to open such a gap is to introduce a periodic potential in the coupled exciton-photon system (see Fig. 4). While periodic potentials for photons are naturally provided by photonic crystals [31], excitonic analogs can be realized, e.g., by applying strain to the quantum well (see Ref. [32] and references therein) or using surface acoustic waves [33,34]. The presence of a periodic potential with period a introduces a Brillouin zone for the polariton spectrum. The coupling of the backfolded bands at the Brillouin-zone boundary then allows to open a gap (see Fig. 5).
To understand the topological nature of such a gap, we recall that most of the winding responsible for the non-trivial topology takes place around the circle of radius q res corresponding to the exciton-photon resonance. For π/a q res , we thus expect the same non-trivial Chern number as in the absence of a periodic potential, while for π/a q res the lowest polariton band should be topologically trivial. To achieve a large topological gap, the Brillouin-zone boundary should correspond to a momentum of the order of (but larger than) q res . Note, however, that one has to be careful if the coupling that opens the gap at the Brillouin-zone boundary itself has some winding. In the exciton-dominated polariton regime (q > q res ), a gap can be opened without introducing any additional winding by using a periodic exciton potential. Purely photonic periodic potentials, on the other hand, can only gap out the excitonic part via the (winding) exciton-photon coupling g q . This results in an additional winding coupling between the excitonic bands w q P± q 2p/a topological bandgap q res Figure 5: Scheme for opening a topological gap using a periodic potential. The solid (dashed) blue and red lines correspond to the dispersion of the lower and upper polaritons with (without) a periodic potential in a simplified one-dimensional scenario. As long as π/a > qres, the lowest polariton band includes the winding around the resonance and the gap opened at the Brillouin-zone boundary is of topological nature.
at the Brillouin-zone boundary which cancels the original winding and makes the gap between the lowest polariton bands topologically trivial. To avoid such complications, we thus focus on periodic exciton potentials. Figure 5 provides an intuitive picture of our gapopening scheme in a simplified one-dimensional case. For the actual two-dimensional Brillouin zone, one essentially recovers this scenario for any cut passing through the center of the Brillouin zone (see Methods). The optimal way to open a topological gap is to maximize the overlap between the gaps obtained along each possible cut. This occurs when the Brillouin-zone geometry is as circular as possible (e.g., hexagonal). Numerical results. To study the edge modes numerically, we start from the same lattice model as in Fig. 3 and set the lattice constant to a/2. We then introduce a periodic potential of period a by alternating the on-site energy between neighboring sites of the square lattice. Figure 6 shows the resulting spectrum and clearly demonstrates the presence of polaritonic edge modes. Since the topological gap ∆ is controlled by the periodic potential, the typical length scale for the edge-mode localization is of the order of a. The size of the gap then determines the edge-mode velocity v ∼ a∆/π. Since the exciton velocity is essentially negligible as compared to the photon velocity c, the edge-mode velocity in turn dictates a ratio v/c between the photon and exciton components of the edge mode. Practical realization. What are the necessary features and parameters for a quantum well to host topolaritons? Most importantly, it must display stable free excitons. In addition, the exciton-photon coupling g q must be as large as possible since it defines, along with the peri-  Figure 6: Topological polariton spectrum with a periodic potential. The spectrum is obtained for a finite-size lattice-version of the Hamiltonian (1) with an additional periodic potential of period a and exhibits in-gap chiral edge modes (colored). The color scale encodes the edge localization, a more intense red(blue) color indicating a larger weight of the wavefunction at the edge at y = 0 (y = L). The system size is 40 × 40 (with lattice constant a = 1) and the other relevant parameters are given by mX = 1·10 5 , mC = 1, ω X 0 = 0.9, g = 0.1,VX = 0.05. odic exciton potential, the relevant gaps in the polariton spectrum. Equation (8) ties g q to the ratio of the (pseudo-)spin-orbit coupling amplitude A to the exciton Bohr radius a 0 , and is valid provided that A/a 0 < M (see also Eq. (5)). Together, the above conditions thus require a large spin-orbit coupling and a bandgap larger than A/a 0 . The size of the topological gap is ultimately limited by the strength of the periodic exciton potential, and must exceed the inverse of the polariton lifetime.
We demonstrate the feasibility of our proposal using alloys of HgTe and CdTe as a guideline, with the spinorbit coupling strength of HgTe and the excitonic parameters of CdTe. For an exciton energy ω X 0 = 0.4eV and a dielectric constant = 10, the condition π/a ∼ q res is fulfilled for a periodic exciton potential with lattice constant a ≈ 500nm. Assuming a spin-orbit coupling amplitude A = 3.365eV Å, an exciton Bohr radius a 0 = 3nm, and a width of 500nm for the photonic waveguide, we then obtain a exciton-photon coupling amplitude |g q | ≈ 6meV using Eq. (8). For an optimal triangular-lattice potential of depth 1meV, a topological gap of 0.2meV is obtained. Such a gap would be large enough to achieve stable polariton edge modes, since typical polariton lifetimes are of the order of tens of picoseconds (1/10ps = 0.07meV) [26]. Larger gaps could be obtained by enhancing the excitonphoton coupling and increasing the strength of the exciton potential. We expect values of the order of 1meV to be within experimental reach.
A promising alternative for realizing topolaritons is provided by monolayers of transition metal dichalcogenides (TMDs). These atomically thin two-dimensional materials with graphene-like (honeycomb) lattice structure have attracted a lot of interest in recent years owing to their coupled spin and valley degrees of freedom as well as their large direct bandgap (1-2eV) and spin-orbit coupling (see, e.g., Ref. [35]). In the presence of an external magnetic field that lifts their valley degeneracy [36][37][38], their electronic properties can be described by an effective two-band Hamiltonian of the form (5) with gap and spin-orbit parameters M and A as large as 0.5-1eV and 3.5-4.5eV Å, respectively [35,39,40]. Exciton-polaritons in the strong-coupling regime have also been achieved, with exciton binding energies close to 1eV and a Bohr radius of the order of 1nm [41]. With such parameters, the size of the topological gap that could be achieved would essentially only be limited by the strength of the periodic exciton potential. For a potential of depth 1meV, a gap of the order of 1meV would be obtained. This would require a magnetic field of about 4-5T to lift the valley degeneracy [36][37][38], corresponding to a regime where the magnetic length is much larger than the exciton Bohr radius, as desired. We remark that TMDs would also provide a promising platform to realize topolaritons with winding number |m| = 2 using circularly polarized light as discussed in the Supplementary Information. In that case the circular polarization of light automatically addresses only one of the two valleys, thus obviating the need for a magnetic field [42][43][44].

Probing topolaritons
The non-trivial topology of topolaritons manifests itself through the existence of chiral edge modes in the excitation spectrum. Probing these modes requires both a means to excite them and the ability to observe them. Due to their mixed exciton-photon nature, chiral edge polaritons can be excited either electronically or optically. Possible electronic injection processes include exciton tunneling [45] or resonant electron tunneling [46]. More commonly, polaritons can be created by addressing their photonic component using a pump laser. Either way, the spectral resolution of the injection process should ideally be smaller than the topological gap in order to single out the chiral edge modes of interest from the rest of the spectrum. A lower spectral resolution can be compensated by better spatial focusing at the edge where the bulk modes have less weight using, e.g., laser spot sizes < a.
Once excited, topological polaritonic edge modes propagate chirally which results in clear differences when observing them upstream and downstream along the edge. Although these chiral edge modes are protected from backscattering, exciton and photon losses translate as a decay of their wavefunction over time. Such dissipation is both harmful and useful: On one hand, the finite decay rate eventually spoils the stability of the edge mode and should not exceed the size of the topological gap for the edge modes to be well-defined. On the other hand, pho- ton losses provide a way to probe the existence of edge modes. For example, photons confined in a waveguide cannot escape when propagating along a smooth edge. Sharp turns (such as corners) would allow to couple to the outside continuum of modes, which could be used to locally detect or inject topological polaritonic edge modes.
To incorporate pumping and loss effects and to complement the lattice-model results of Fig. 6, we simulated what we envision as a typical probe experiment using a driven-dissipative Gross-Pitaevskii equation based on the continuum model of our proposal (see Supplementary Information). We considered a scenario where a continuous-wave pump laser with a frequency ω p set within the topological gap is switched on at some initial time t = 0 and focused onto a spot of diameter ≈ a at the edge. Figure 7 shows the time-evolved exciton and photon fields obtained in that case. The photonic part of the edge modes can in principle be easily retrieved by detecting the light leaking out of the system. For example, the photon component shown in Fig. 7 would be observed in a system with losses that are uniform and directed in the z-direction. We remark that exciton-exciton interactions are expected to be present in typical experimental realizations (see, e.g., [26]). Our numerical studies indicate that the edge modes remain stable in the presence of weak interactions (i.e., for interactions leading to a blueshift smaller than the topological gap), as expected from their chiral nature (see Supplementary Information and Video).

Conclusions
In this manuscript, we have shown that topological polaritons ("topolaritons") can be created by combining ordinary photons and excitons. The key ingredients of our scheme are a winding exciton-photon coupling and a suitable exciton potential to open a topological gap. Promising candidates for the realization of the winding coupling are two-dimensional photonic modes coupled to semiconductor quantum wells or monolayer transition metal dichalcogenides, with spin-orbit interaction and a finite Zeeman field. Although combining all necessary ingredients may be experimentally challenging, we emphasize that all underlying requirements have already been achieved. Realizing topolaritons thus seems within experimental reach. In light of the importance of a Brillouin zone for the opening of the gap, it would be interesting to study whether our proposal can be extended to systems known to exhibit strong polariton potentials, such as lattices of coupled micropillars [47].
Topolaritons manifest themselves through chiral edge modes which are, by definition, protected against backscattering. The directionality of these one-way channels can in principle easily be reversed, either by modifying the Zeeman field or by addressing their time-reversed partners appearing at different energies. This makes for a versatile platform for the directed propagation of excitons and photons, with the possibility of converting between the two. An important feature of topolaritons is their strong excitonic component, which allows for interactions to come into play. Interaction effects are notoriously hard to achieve in purely photonic topological systems [32]. Here they could provide an alternative and more flexible way to realize the periodic exciton potential required in our proposal. Indeed, one may envision to create an effective potential by injecting a different exciton species with a periodic density profile [48]. Interactions may also lead to novel avenues towards the observation of other intriguing topological phenomena such as non-equilibrium fractional quantum Hall effects [8,49,50].
More broadly, our route to obtaining topological polaritons reveals a generic approach for achieving nontrivial topological states by mixing trivial bosonic components. We expect extensions of this idea to other physical systems to lead to the realization of yet more surprising bosonic topological phenomena.

Methods
Derivation of the winding exciton-photon coupling. Applying the minimal coupling substitution k → k + eÂ to the second-quantized representation of the quantum-well Hamiltonian (5) leads to an electron-photon coupling of the form where we have focused on the region around the exciton-photon resonance. Note that we have replaced the electron operators corresponding to the original angular-momentum basis [see Eq. (5)] by conduction-and valence-band operators. Both basis are related by a spin-orbit induced rotation of the order of Ak/M and are equivalent under the assumption that A/M is small as compared to the Bohr radius a 0 ∼ 1/k of the excitons. We have verified numerically that the winding remains unchanged for larger A/(M a 0 ) ∼ 1 (for excitons with s-wave character). In that case, however, the amplitude of the exciton-photon coupling becomes suppressed.
To express the electron-photon coupling in terms of excitons, we invert Eq. (6) as c † c,k+q/2 c v,k−q/2 = n φ * n (k)b (n) † q . Focusing on energies close to the lowest-lying exciton allows for dropping terms including higher exciton modes with n > 1. With the two-dimensional Fourier transform of the s-wave wavefunction φ 1 (k) = 2 √ 2πa 0 1 + (ka 0 ) 2 −3/2 , we then obtain Eq. (8) of the main text. Note that the apparent singularity of Eq. (8) at q = 0 is an artifact of the large in-plane momentum limit. For a small in-plane momentum component, the TM-polarized modes can no longer be neglected. In fact, the TE and TM polarization modes have equally large in-plane electric fields if the momentum points predominately in the z-direction. In this regime the direction of the electric field is mainly determined by the polarization of the photons and only weakly dependent on the in-plane momentum, which spoils the winding nature of the coupling. This small in-plane momentum regime is characterized by a quadratic photon dispersion. For Eq. 8 to be valid, we thus assume that the photon dispersion is linear in the region around resonance (see Fig. 2).
Topological gap opening for a two-dimensional Brillouin zone. Assuming a square Brillouin zone, a gap of size V X opens at the point X = (π/a, 0) similarly as in Fig. 5. The main deviation from the one-dimensional case is the finite energy difference between the polaritons at the points X and M = (π/a, π/a). For a global gap, V X should be larger than ω P − M − ω P − X . Note that V X also broadens the exciton bands that are backfolded into the first Brillouin zone from regions of the spectrum far away from resonance. These backfolded excitons are in an energy range ω X 0 ± 2V X . There is therefore a tradeoff between the large exciton-photon coupling required to separate the lowest polariton band from the backfolded excitons (large g 2 M /(ω C M − ω X 0 ) as compared to V X ) and the large V X required to open a global gap. To satisfy these requirements in the best possible way, the Brillouin zone should be as circular as possible (e.g., hexagonal) to minimize the energy difference between the points located at its boundary. For an optimal choice of a, V X , and Brillouin-zone geometry, we expect the size of the gap to be determined by the smallest of the two values V X and g 2 M /(ω C M − ω X 0 ).