Resonant laser excitation and time-domain imaging of chiral topological polariton edge states

We investigate the dynamics of chiral edge states in topological polariton systems under laser driving. Using a model system comprised of topolgically trivial excitons and photons with a chiral coupling proposed by Karzig et al.[Phys. Rev. X 5, 031001 (2015)], we investigate the real-time dynamics of a lattice version of this model driven by a laser pulse. By analyzing the time- and momentum-resolved spectral function, measured by time- and angle-resolved photoluminescence in analogy with time- and angle-resolved photoemission spectroscopy in electronic systems, we find that polaritonic states in a ribbon geometry are selectively excited via their resonance with the pump laser photon frequency. This selective excitation mechanism is independent of the necessity of strong laser pumping and polariton condensation. Our work highlights the potential of time-resolved spectroscopy as a complementary tool to real-space imaging for the investigation of topological edge state engineering in devices.

Here we show by calculations for a topological polariton lattice model how real-space imaging and timeresolved spectroscopy go hand-in-hand in demonstrating selective edge mode excitations by laser driving. We demonstrate that selective excitation of chiral edge modes is possible by laser pumping without the necessity of polariton condensation due to nonlinearities (i.e., without polariton-polariton interactions) if the pump laser frequency is tuned in resonance with the edge mode. A complementary real-space imaging reveals the edge localization of the populated modes.
Our setup is shown in Fig. 1. A quantum well (QW) harbors exciton modes that couple via dipole-dipole in- teractions to photon modes in a cavity, leading to the formation of exciton polaritons. As suggested for instance in Ref. 53 the QW could just as well be replaced by monolayer transition metal dichalcogenides 57 . We will assume that there is a chiral coupling between excitons and polaritons 53 .
For our calculations we start from a tight-binding model on a two-dimensional square lattice in a ribbon geometry, as shown in Fig. 2. Consider a lattice L of size N = N x × N y with two bosonic modes at each site, representing photon and exciton populations, respectively. In the following,â 0 i (â 1 i ) denotes the photonic (excitonic) field operator at site i. Our model is a generalization of the Qi-Wu-Zhang model 58 , also called half Bernevig-Hughes-Zhang model 59,60 . We note that bosonic transport in a variation of this model including a non-linear interaction term has been studied by Weiß 61 where i+d ν is the index of the nearest neighbor of site i in positive ν ∈ {x, y} direction. The hopping is determined by the matrices t(0) = diag(−ε 0 , −ε 1 ) and in order forĤ 0 to be Hermitian. The parameters t α are the photon and exciton hopping amplitudes, ε α a constant energy offset, and g 0 the exciton-photon coupling strength. Assuming periodic boundary conditions in both x and y direction, H 0 can be written aŝ Here, L denotes the discretized first Brillouin zone of the lattice, is the tight-binding dispersion, and is the momentum-dependent exciton-photon coupling where x and y denote the lattice spacing in the respective direction. The operatorsâ α are obtained from the field operators by a discrete Fourier transform. The static Hamiltonian has a band structure that corresponds to a tight-binding approximation of the continuum model presented by Karzig et al. 53 . The momentum-dependence of the coupling g(k) leads to a non-zero Chern number C ± = ∓1 of the upper (+) and lower (−) polariton band. The time-dependent driving is implemented by the operator Here, for simplicity, the driving term acts directly only on the photonic mode α = 0 at a single site i 0 which is located on the open side of the lattice boundary.
In order to show the presence of topological edge states in the system, we now add cylindrical boundary conditions, with periodic boundary conditions in x direction and open boundary conditions in y direction (Fig. 2a). The corresponding band structure obtained by a partial Fourier transform in x direction is shown in Fig. 2b.
Our simulations are performed in the semi-classical limit, where the bosonic operators are replaced by scalar complex fields. This corresponds to restricting the possible quantum states to the coherent states |ψ parametrized by the scalar complex field ψ ∈ C 2N , which satisfiesâ α i |ψ = ψ α i |ψ . Since the Hamiltonian (1), including the driving termF (t), is of second order in the creation and annihilation operators, the coherent states are closed under the corresponding time evolution. Thus, if started in a coherent initial state, expectation values obtained from the semi-classical approach are exact. The equations of motion for the fields ψ α i are ( = 1) for (i, α) = (i 0 , 0). The initial state is chosen to be the coherent polaritonic vacuum state with ψ α i (0) = 0 for all i and α.
In the semiclassical approximation, the double-time lesser Green's functions can be obtained directly from the fields as From the Green's functions, we compute the timeresolved spectral density 62 I(ω, k) = dt 1 dt 2 S(t 1 )S(t 2 )e iω(t1−t2)G< (k; t 1 , t 2 ) ix,y,α;ix,y,α (t 1 , t 2 )] (11) which is Fourier-transformed along the periodic ribbon direction x and traced over y direction. We also trace here over the photon and exciton index, i.e., we compute the exciton-polariton spectral density. In principle, it is easily possible to resolve the spectral contributions of excitons and photons separately, but this will not be crucial for our analysis of edge localization of pumped modes below. In Fig. 3 we present the time-and momentum-resolved spectral functions, mimicking time-domain photoluminescence spectra, of a continuously irradiated ribbon of size N x × N y = 256 × 8 with laser focused to a lattice site i 0 at the lower edge (y = 0) of the ribbon. The driving frequency of the external laser is varied to be below [ Fig. 3(a-d)] the topological band gap, within the gap region [ Fig. 3(e)], and above the gap [ Fig. 3(g-h)]. As can be seen from the time-resolved spectral density, the polariton branches are selectively occupied by the resonant laser excitation, which shows that the external driving frequency is the main tuning knob for populating polari-ton branches in the absence of polariton-polariton interactions and associated mechanisms for polariton condensation.
Finally, we analyze in Fig. 4 the edge localization of the light-induced states through real-space analysis of the light-induced populations. Fig. 4(a) shows the localization at the lower edge, i.e., the ratio of intensity at y = 1 integrated (along the x direction) and the total intensity in the system at a given point in time. A clearly resonant resonant behavior is observed when the driving frequency matches the edge state energy of ≈ 1.8|t C |. The degree of localization is even more pronounced for the excitonic component of the polariton wavefunction. In Fig. 4(b) we analyze the localization integrated over the lower half 1 ≤ y ≤ 4 of the ribbon, which consistently shows same the resonant behavior. In Fig. 4(c) we show representative real-space images for the photon and exciton densities in the on-resonance case. Here, besides the already discussed localization at the lower edge, one can also observe the chiral propagation of the edge mode from left to right as indicated by the arrow. By clear contrast, in Fig. 4(d) both the edge selectivity and chirality of the laser-pumped polariton populations are absent. These real-space images thus provide complementary information to the spectroscopic pictures presented in Fig. 3.
In summary we have shown lattice model simulations for the temporally and spatially resolved dynamics of chiral topological exciton polaritons. We have demonstrated the selective excitation of chiral edge modes provided that the external driving frequency is sufficiently closely tuned to the topological band gap. An interesting open question pertains to the role of polariton-polariton interactions and the question of the importance of condensation versus resonant excitation for the experimentally observed chiral edge modes 56 .
As an upshot from our calculations, it is straightforward to extend the formalism employed here to compute time-resolved spectroscopy in periodically driven manybody systems, both with continuous and ultrashort laser pulses. Specifically for continuous driving this is the realm of Floquet engineering 63-66 of effective band structures, which is by itself a rapidly growing research field. The interface between the different fields of quantum simulators, nonequilibrium quantum materials science, and polaritonic condensates promises many interesting applications for future quantum technologies.