Microscopic description of exciton-polaritons in microcavities

We investigate the microscopic description of exciton-polaritons that involves electrons, holes and photons within a two-dimensional microcavity. We show that in order to recover the simplified exciton-photon model that is typically used to describe polaritons, one must correctly define the exciton-photon detuning and exciton-photon (Rabi) coupling in terms of the bare microscopic parameters. For the case of unscreened Coulomb interactions, we find that the exciton-photon detuning is strongly shifted from its bare value in a manner akin to renormalization in quantum electrodynamics. Within the renormalized theory, we exactly solve the problem of a single exciton-polariton for the first time and obtain the full spectral response of the microcavity. In particular, we find that the electron-hole wave function of the polariton can be significantly modified by the very strong Rabi couplings achieved in current experiments. Our microscopic approach furthermore allows us to obtain the effective interaction between identical polaritons for any light-matter coupling. Crucially, we show that the standard treatment of polariton-polariton interactions in the very strong coupling regime is incorrect, since it neglects the light-induced modification of the exciton size and thus greatly overestimates the effect of Pauli exclusion on the Rabi coupling, i.e., the saturation of exciton oscillator strength. Our findings thus provide the foundations for understanding and characterizing exciton-polariton systems across the whole range of polariton densities.

Figure 1

Feynman diagrams involved in the renormalization of the cavity photon frequency. (a) Dyson equation for the photon propagator (double wavy line) in terms of the bare photon propagator (single wavy line) and the photon self-energy (shaded ellipse). The black circles indicate the light-matter coupling $g$. (b) Separation of the self-energy into the bare electron-hole polarization bubble (left) and a term containing any positive number of repeated Coulomb interactions (right). The lines represent bare electrons and holes, and the square represents the electron-hole $T$ matrix. (c) The equation satisfied by the $T$ matrix, where the dotted wavy line represents the Coulomb interaction.

Figure 2

Energies (top) and photon fractions (bottom) of the lower and upper polaritons obtained within our microscopic model in Eq. (11) (blue solid lines) and within the two-level model (18) (purple dashed lines). We show the results as a function of detuning, and from left to right we consider increasing strength of the Rabi coupling: [(a) and (d)] $\mathrm{\Omega }/E_{\mathrm{B}}=0.1$, [(b) and (e)] 0.5, and [(c) and (f)] 1. In (a)–(c), the horizontal dashed lines are the $1s$ and $2s$ exciton states with energies ${\varepsilon }_1=-E_{\mathrm{B}}$ and ${\varepsilon }_2=-E_{\mathrm{B}}/9$, respectively, while the shaded regions correspond to the electron-hole continuum.

Figure 3

Transmission spectrum $\mathcal{T}(\mathbf{Q},E)$ of a single photon at detuning $\delta =0$ and for Rabi couplings (a) $\mathrm{\Omega }/E_{\mathrm{B}}=0.1$, (b) 0.5, and (c) 1. The polariton dispersions predicted from the two-level model in Eq. (A5) are shown as white dashed lines, while the bare photon and $1s$ and $2s$ exciton dispersions are shown as dashed black lines. The masses are taken to be the realistic values [44]: $m_e=0.063m_0,m_h=0.5m_0,\text{and}{m}_c={10}^{-4}{m}_0$, with $m_0$ the free electron mass. To enhance visibility over the large energy range displayed in the figures, we are taking the photon lifetime broadening to be $\mathrm{\Gamma }=0.1E_{\mathrm{B}}$, which exceeds typical values in high-quality-factor microcavities. Each spectrum has been normalized to the maximum transmission.

Figure 4

[(a)–(d)] Normalized electron-hole wave function as a function of separation and [(e) and (f)] the mean electron-hole separation as a function of detuning. The upper (lower) panels correspond to the upper (lower) polariton, and we take $\delta /E_{\mathrm{B}}=-0.5$ in (a) and (c) and $\delta /E_{\mathrm{B}}=0.5$ in (b) and (d). We show the exact results obtained from our microscopic theory (blue lines) for the Rabi coupling strengths of $\mathrm{\Omega }/E_{\mathrm{B}}=0.1$ (dotted line), 0.5 (dashed line), and 1 (solid line). Note that the numerically calculated wave functions in (a)–(d) are only shown for $r>0.005a_0$. We also show the $1s$ exciton wave function and average separation (thin black line). In (e) and (f), the purple lines correspond to the result (27) from the variational approach [18, 31], where we take the detuning to be $\tilde{\delta }$ (see main text).

Figure 5

Polariton-polariton interaction strength $g_{\mathrm{PP}}$ for several values of the light-matter coupling: (a) $\mathrm{\Omega }/E_{\mathrm{B}}=0.1$; (b) $\mathrm{\Omega }/E_{\mathrm{B}}=0.5$; and (c) $\mathrm{\Omega }/E_{\mathrm{B}}=1$. We compare our numerical results obtained from Eq. (31) (blue solid lines) with the simple exciton approximation $g_{\mathrm{PP}}^{\left(0\right)}={\beta }^4{g}_{\mathrm{XX}}$ (purple dashed lines) and the commonly accepted interaction strength that includes the exciton oscillator strength saturation correction (C10): $g_{\mathrm{PP}}^{\mathrm{sat}}=g_{\mathrm{PP}}^{\left(0\right)}+\frac{32\pi }{7}\mathrm{\Omega }{a}_0^2{\beta }^3\left|\gamma \right|$ (gray dash-dotted line).