Molecular and solid-state topological polaritons induced by population imbalance

Abstract

Strong coupling between electronic excitations in materials and photon modes results in the formation of polaritons, which display larger nonlinearities than their photonic counterparts due to their material component. We theoretically investigate how to optically control the topological properties of molecular and solid-state exciton-polariton systems by exploiting one such nonlinearity: saturation of electronic transitions. We demonstrate modification of the Berry curvature of three different materials when placed within a Fabry-Perot cavity and pumped with circularly polarized light, illustrating the broad applicability of our scheme. Importantly, while optical pumping leads to nonzero Chern invariants, unidirectional edge states do not emerge in our system as the bulk-boundary correspondence is not applicable. This work demonstrates a versatile approach to control topological properties of novel optoelectronic materials.

Keywords: exciton–polariton; strong light–matter coupling; topological polaritons.

Figure 1:

Illustration of the system under study. Porphyrin (molecules at the center) and perylene (green blocks) placed within a Fabry–Perot cavity and pumped with circularly polarized light.

Figure 2:

Three-level model of a metalloporphyrin molecule. (a) Illustration of circularly polarized light exciting a metalloporphyrin molecule. (b) Three-level model of porphyrin with a ground state $\left|\mathrm{G}\right.⟩$ and two degenerate excited states $\left|{+}_{\text{mol}}\right.⟩,\left|{-}_{\text{mol}}\right.⟩$ . The transition dipole moment for a transition from $\left|\mathrm{G}\right.⟩$ to $\left|{\pm }_{\text{mol}}\right.⟩$ is ${\mathit{\mu }}_{\pm }={\mu }_0(\widehat{\mathbf{x}}\pm i\widehat{\mathbf{y}})/\sqrt{2}$ . The number of yellow circles at each state represents the fraction of molecules in that state. Here, the ratio of the fraction of molecules in the ground, f _G, and $\left|{\pm }_{\text{mol}}\right.⟩$ excited states, f _±, is f _G : f ₊ : f ₋ = 3 : 1 : 0. Such population ratios can be achieved through pumping with circularly polarized light.

Figure 3:

Berry curvature and degree of circular polarization of the bands. (a–d) Berry curvature of the lowest energy band, Ω₁(k), and (e–h) a slice of the band structure at k _y = 0 of the lower two bands, under different levels of optical pumping, which create populations: (a, e) f ₊ = f ₋ = 0, (b, f) f ₊ = 0.3, f ₋ = 0, (c, g) f ₊ = 0, f ₋ = 0.3, and (d, h) f ₊ = f ₋ = 0.3. (e–h) The colors of the band indicate the value of the Stokes parameter, S ₃(k), which measures the degree of circular polarization of a mode (Eq. (8)). The Chern numbers C ₁ and C ₂ of the bands are also specified and are nonzero under time-reversal symmetry (TRS) breaking, that is, when f ₊ ≠ f ₋. We used parameters β ₀ = 0.1 eV, β = 9 × 10⁻⁴ eV μm², ζ = 2.5 × 10⁻³ eV μm, m* = 125ℏ ² eV⁻¹ μm⁻², E ₀ = 3.80 eV, and ℏω _e = 3.81 eV (see Section S4 in Supporting Information for details).

Figure 4:

The Stokes parameter, S ₃(k), which is a measure of the degree of circular polarization of a mode (Eq. (8)), under pumping with (a, c) σ ₊ polarized light, which creates populations f ₊ = 0.3, f ₋ = 0, and (b, d) σ ₋ polarized light, which creates populations f ₊ = 0, f ₋ = 0.3 of the two lowest energy bands (band 1 and 2 as indicated in the inset). We used parameters β ₀ = 0.1 eV, β = 9 × 10⁻⁴ eV μm², ζ = 2.5 × 10⁻³ eV μm, m* = 125ℏ ² eV⁻¹ μm⁻², E ₀ = 3.80 eV, and ℏω _e = 3.81 eV (see Section S4 in Supporting Information for details).

Figure 5:

Solid-state polariton systems where population imbalance induces non-trivial topology. (a) Illustration of Ce:YAG (salmon block) and perylene (green blocks) within a Fabry–Perot cavity. (b) Atomic levels of Ce³⁺ ions embedded in yttrium aluminum garnet (YAG) where the yellow circles indicate the fraction f _↓ of Ce³⁺ ions in the $\left|4f(1)\downarrow \right.⟩$ state and the fraction f _↑ in the $\left|4f(1)\uparrow \right.⟩$ state after optical pumping. The transition dipoles ${\mathit{\mu }}_{\pm }={\mu }_0(\widehat{\mathbf{x}}\pm i\widehat{\mathbf{y}})/\sqrt{2}$ are also indicated. (c) Berry curvature of the lowest energy band, Ω₁(k), under pumping with circularly polarized which creates populations f _↓ = 0.4 and f _↑ = 0.6. (d) Illustration of monolayer MoS₂ and perylene (green blocks) within a Fabry–Perot cavity. (e) Illustration of A-excitons in the K and K′ valleys of monolayer MoS₂. (f) Berry curvature of the lowest energy band, Ω₁(k), under pumping with circularly polarized which creates exciton populations f _K = 0.3 and f _K′ = 0. We used parameters β ₀ = 0.1 eV, β = 9 × 10⁻⁴ eV μm², ζ = 2.5 × 10⁻³ eV μm, m* = 125ℏ ² eV⁻¹ μm⁻², (c) E ₀ = 2.50 eV, ℏω _e = 2.53 eV and (f) E ₀ = 1.80 eV, ℏω _e = 1.855 eV (see Section S4 in Supporting Information for details).