Light-enhanced dipolar interactions between exciton polaritons

Abstract

We consider the scenario of excitons in a semiconductor bilayer that are strongly coupled to cavity photons, leading to the formation of dipolar exciton polaritons (dipolaritons). Using a realistic pseudopotential for the dipolar interactions, we exactly determine the scattering between dipolaritons, accounting for the hybridization between interlayer and intralayer excitons. Similar to conventional non-dipolar polaritons, we find that the light-matter coupling enhances the interactions between dipolaritons by forcing excitons to scatter at energies that would otherwise be forbidden in ordinary exciton-exciton collisions. However, we show that this light enhancement is larger for long-range dipolar interactions than for short-range intralayer interactions, and is sensitive to the (non-uniform) dielectric environment of the bilayer. Crucially, we find that the largest dipolariton interactions are achieved for transition metal dichalcogenide bilayers in vacuum. Our results thus reveal the optimal dipolariton setup for realizing strong photon correlations.

Figures (6)

• Figure 1: (a) Schematic of the dipolariton in a bilayer. Direct excitons DX$_1$ and DX$_2$ are coupled to the cavity photon (yellow region), while each DX hybridizes with the corresponding indirect exciton IX via tunneling $t$ of the hole (red). The IX dipole moment depends on the interlayer separation $d$ and the dielectric constant $\kappa$ of the surrounding environment. For co-polarized photons, DX$_1$ is in the $K'$ valley and DX$_2$ is in the $K$ valley (see text). (b) Photon spectral function, featuring the five polariton branches. We also show the photon (C) and hybrid excitons (hX$^+$ and hX$^-$) in the absence of light-matter coupling (dashed white). We use parameters inspired by MoS$_2$ homobilayer experiments Leisgang2020Lorchat2021Datta2022Louca2023: $\Omega/\varepsilon_\text{X} = 0.085$, $t/\varepsilon_\text{X} = 0.17$, $\delta_\text{C}/\varepsilon_\text{X} = -0.34$, $\delta_\text{IX}/\varepsilon_\text{X} = -0.17$, $\Delta/\varepsilon_\text{X} = 0.085$, and $\Gamma/\varepsilon_\text{X} = 0.01$, where the reference scales $a_\text{X}$ and $\varepsilon_\text{X}$ correspond to $\kappa=1$ (see text).
• Figure 2: (a) Schematic of the intravalley DX-DX(IX) (left) and IX-IX (right) interaction potentials, where the latter exhibits a long-range dipolar tail. (b) Exciton $T$ matrices at zero momentum and zero tunneling as a function of collision energy, i.e., the energy measured from the corresponding two-particle continuum. The blue and red lines show the IX-IX $T$ matrix $T_{33}=T_{44}$ for vacuum ($\kappa=1$) and hBN ($\kappa=3.76$), respectively. The yellow and green lines show the DX-DX $T$ matrix $T_{11}=T_{22}$ for $\kappa=1$ and $\kappa=3.76$, respectively. The gray dot-dashed line is the 2D universal low-energy expression \ref{['eq:2D T-matrix']} expected for DX-DX interactions, with $a_\text{2D}/a_\text{X}=0.39$ obtained from a fit at small collision energy. In panel (b), we use $\rho_{0}/a_{0} = 40$ and $d/a_{0} = 6$, with the reference scales $a_\text{X}$, $\rho_{0}$, $a_{0}$, and $\varepsilon_\text{X}$ fixed by taking $\kappa=1$.
• Figure 3: (a) LP-LP interaction constant at zero momentum as a function of photon detuning, where $\epsilon^\text{X}_\mathbf{0}=E^{-}_{\mathbf{0},1}$ for dipolaritons and $\epsilon^\text{X}_\mathbf{0}=0$ for conventional polaritons. The blue and red lines correspond to dipolaritons for vacuum ($\kappa=1$) and hBN ($\kappa=3.76$), respectively, with $\Delta/\Omega = 1$. The black dashed lines show the corresponding off-shell approximations. The limit $\delta_\text{IX}\to+\infty$ (yellow and green lines) corresponds to conventional polaritons. (b,c) Density plots of the LP-LP interaction constant for $\kappa=1$ and $\kappa=3.76$, respectively. The additional black dashed line indicates $\delta_\text{C} = E^{-}_{\mathbf{0},1}$. In all panels, we use $\rho_{0}/a_{0} = 40$, $d/a_{0} = 6$, $\Omega/\varepsilon_\text{X} = 0.085$, $t/\varepsilon_\text{X} = 0.17$, and we use $\delta_\text{IX}/\varepsilon_\text{X} = -0.17$ unless otherwise specified. The reference scales $\rho_{0}$, $a_{0}$, and $\varepsilon_\text{X}$ correspond to $\kappa=1$.
• Figure S1: (a) Energy of the ground-state DX (blue) and IX (red) as a function of dielectric constant. We measure the energy from the corresponding electron-hole continuum. (b) Interaction constant for intravalley DX-DX (blue), IX-IX (red), and DX-IX (yellow) scattering obtained from the Born approximation. We suppress the index $\eta$ for simplicity. (c) Effective size of the exciton $a_\text{X}$ for short-range interactions, and short-distance cutoff $r_{0}$ for dipolar interactions. In all panels, we take equal masses for the electron and hole, and we use $\rho_{0}/a_{0} = 40$ and $d/a_{0} = 6$, with the reference scales $\rho_{0}$, $a_{0}$, and $\varepsilon_\text{X}$ fixed by taking $\kappa=1$.
• Figure S2: (a,b,c) LP-LP interaction constant at zero momentum as a function of Stark shift. The blue and red lines, respectively, correspond to the case for vacuum ($\kappa=1$) and hBN ($\kappa=3.76$). (d,e,f) Corresponding Hopfield coefficients at zero momentum. The yellow, solid (dashed) red, and solid (dashed) blue lines, respectively, correspond to the photon, DX$_{1}$ (DX$_{2}$), and IX$_{1}$ (IX$_{2}$) fractions. Each column corresponds to a different value of the photon detuning, $\delta_\text{C}/\Omega = -3,-2.5,-2$ from left to right. In all cases, we use $\rho_{0}/a_{0} = 40$, $d/a_{0} = 6$, $\Omega/\varepsilon_\text{X} = 0.085$, $t/\varepsilon_\text{X} = 0.17$, and $\delta_\text{IX}/\varepsilon_\text{X} = -0.17$, where the reference scales $\rho_{0}$, $a_{0}$, and $\varepsilon_\text{X}$ correspond to $\kappa=1$.