Theory predicts 2D chiral polaritons based on achiral Fabry-Pérot cavities using apparent circular dichroism

Abstract

Realizing polariton states with high levels of chirality offers exciting prospects for quantum information, sensing, and lasing applications. Such chirality must emanate from either the involved optical resonators or the quantum emitters. Here, we theoretically demonstrate a rare opportunity for realizing polaritons with so-called 2D chirality by strong coupling of the optical modes of (high finesse) achiral Fabry-Pérot cavities with samples exhibiting "apparent circular dichroism" (ACD). ACD is a phenomenon resulting from an interference between linear birefringence and dichroic interactions. By introducing a quantum electrodynamical theory of ACD, we identify the design rules based on which 2D chiral polaritons can be produced, and their chirality can be optimized.

Fig. 1. Schematic depiction of apparent circular dichroism (ACD) and its implementation within an achiral Fabry–Pérot (FP) cavity.

Energy level diagram (a) and schematic of the transition dipoles (b) of a minimal ACD sample involving two bright, nondegenerate, and oblique transition dipoles. Angular transition frequencies ω₁ and ω₂, transition dipoles μ₁ and μ₂, and inter-dipole angle β₁₂ are indicated. Shown below is a schematic depiction of ACD due to linear birefringence (LB) and linear dichroism (LD) resulting from forward (c) and backward (d) propagation, yielding absorption of left-handed (green) and right-handed (orange) circularly-polarized light, respectively. The prime on LD indicates a 45° rotation in the plane of polarization. Also shown is a depiction of the interactions between an ACD sample and an achiral FP cavity (e) involving the selective absorption of the λ = + polarization mode.

Fig. 2. Survey of 2D chiral interaction terms.

2D chiral interaction terms σ_n for two transition dipoles as a function of cavity frequency Ω (a), transition dipole moment ratio μ₂: μ₁ with μ₁ fixed (b), damping parameter γ (c), and transition frequency gap ω₂ − ω₁ with ω₁ fixed (d). Shown are results from a second-order Mueller calculus treatment, using Eq. (11) (dotted curve), alongside those from an infinite-order treatment (solid curve; see text), with ω₁ = 2.0 eV ℏ⁻¹ and ω₂ = 3.0 eV ℏ⁻¹ [except for (d)], while the angle between transition dipoles is set to β₂₁ = 45°. Unless otherwise noted, γ = 0.10, μ₂: μ₁ = 1.0, and Ω = 2.1 eV ℏ⁻¹ in order to be slightly off-resonant with ω₁. Other material parameters are set to μ₁ = 10.0 D, v = 4.5 nm³, and ϵ_∞ = 8.0, with v being the unit cell volume of the involved crystal and with ϵ_∞ being its effective high-frequency dielectric constant. The orange vertical line indicates points with the same parametrization across the different panels.

Fig. 3. Polaritonic dissymmetry g~α as a function of (angular) cavity frequency Ω for the minimal configuration of two transition dipoles.

Shown are results for a vector potential A₀ = 35 eV e⁻¹c⁻¹ (a, b) and for A₀ = 70 eV e⁻¹c⁻¹ (c, d). Depictions of ${\tilde{g}}^{\alpha }$ are separated over two Panels to avoid overlap, and in each Panel corresponds to the polariton dispersions indicated by the solid curves (other polariton dispersions are indicated by the dashed curves as a reference). Results are obtained by numerical diagonalization of the Hamiltonian given by Eq. (13), with inter-dipole angle β₂₁ = 45°, (angular) transition frequencies ω₁ = 2.0 eV ℏ⁻¹ and ω₂ = 3.0 eV ℏ⁻¹, transition dipole moments μ₁ = μ₂ = 10.0 D, crystal unit cell volume v = 4.5 nm³, and high-frequency dielectric constant ϵ_∞ = 8.0.

Fig. 4. Polaritonic dissymmetry for the lowest-energy polariton, g~α=1.

Shown are results as a function of (angular) cavity frequency Ω (a), transition dipole moment ratio μ₂: μ₁ (b), damping factor γ (c) and (angular) transition frequency difference ω₂ − ω₁ (d). Unless noted otherwise, A₀ = 35 eV e⁻¹c⁻¹ and all other parameters are identical to Fig. 2, including an inter-dipole angle β₂₁ = 45°, a crystal unit cell volume v = 4.5 nm³, and an effective high-frequency dielectric constant ϵ_∞ = 8.0. The orange vertical line indicates points with the same parametrization across the different panels.

Fig. 5. Polaritonic dissymmetry and dispersion with and without the three-state approximation (TSA).

Polaritonic dissymmetry ${\tilde{g}}^{\alpha }$ as a function of (angular) cavity frequency Ω for the minimal configuration of two transition dipoles obtained through a numerical diagonalization of the Hamiltonian given by Eq. (13) (a), and comparative results within the TSA (b) which only includes explicitly the lowest-energy excited state (n = 1). Unless noted otherwise, parameters are identical to Fig. 2, while the vector potential is taken to be A₀ = 35 eV e⁻¹c⁻¹. Note that colors have been linearly interpolated for a smoother gradient.

Fig. 6. 2D chiral polaritons in BDT-based oligothiophene.

Polaritonic dissymmetry ${\tilde{g}}^{\alpha }$ for BDT-based oligothiophene as predicted by our theory (a). Overlapping states drawn in order of $\mid {\tilde{g}}^{\alpha }\mid$ such that states with greatest $\mid {\tilde{g}}^{\alpha }\mid$ are visible. The molecular model of the BDT-based oligothiophene was adopted from our previous work (see text for details). Molecular structure shown in (b) with relevant ground-to-excited state transition dipoles indicated, labeled 1, 2, and 3, in order of increasing transition energy (orientation of dipoles relative to the molecular structure is arbitrary to within variations between electronic structure calculations and spectral fitting). Also shown are results without the vibrational modes and only including transitions 1 and 3 (c, d).