Nature and dynamics of low-energy exciton polaritons in semiconductor microcavities

Low-energy polaritons in semiconductor microcavities are important for many processes, such as, e.g., polariton condensation. Organic microcavities frequently feature both strong exciton-photon coupling and substantial scattering in the exciton subsystem. Low-energy polaritons possessing small or vanishing group velocities are especially susceptible to the effects of such scattering that can render them strongly localized. We compare the time evolution of low-energy wave packets in perfect microcavities and in a model one-dimensional cavity with diagonal disorder to illustrate this localization of polaritons and to draw attention to the need to explore its consequences for the kinetics and collective properties of polaritons.

Figure 1

The energy spectrum of the polaritonic eigenstates in a 1D model microcavity described by the Hamiltonian (10) with $N=1500$ molecular sites and $N=1500$ photon modes. Parameters of the systems are as follows: average exciton energy $\epsilon =2\mathrm{eV}$, cavity photon cutoff energy $\Delta =1.9\mathrm{eV}$, dielectric constant $ϵ=3$, and exciton-photon interaction energy $\gamma =0.15\mathrm{eV}$. The dashed lines show the energy eigenvalues in the system without exciton disorder, $\sigma =0$; the solid lines correspond to the spectrum in the system with disorder, $\sigma =0.03\mathrm{eV}$. Here the energies are shown as a function of state “number” sorted in an increasing energy order, separately for the lower (LP) and upper (UP) polariton branches. In the system without disorder, the state numbers would be immediately convertible to the corresponding wave vectors. On this scale, the UP branches of perfect and disordered systems are hardly distinguishable.

Figure 2

Examples of the spatial structure of the photon part ${\mid {\Psi }_p\mid }^2$ of the polariton eigenstates in a 1D microcavity with disorder. (a) Four states, shown by different lines, from the very bottom of the LP polariton branch with energies within the range of $1.76–1.77\mathrm{eV}$ (see the spectrum in Fig. 1). The inset shows one of these states in more detail. Dots in the inset correspond to the sites of the underlying lattice. (b) One state with a higher energy close to $1.82\mathrm{eV}$. The inset shows part of the spatial structure of this state in more detail. Dots correspond to the lattice sites and spatial oscillations of the wave function are clearly seen.

Figure 3

Comparison of the spatial structure of the (a) photon ${\mid {\Psi }_p\mid }^2$ and (b) exciton ${\mid {\Psi }_e\mid }^2$ components of one of the localized polariton eigenstates near the bottom of the LP branch. Wave functions are shown by the thick lines with dots corresponding to the lattice sites. Thin lines in the top parts of both panels display arbitrarily scaled diagonal disorder energies along with their base line shown with dashes. The diagonal disorder shown in panel (b) is the original uncorrelated site disorder of Eq. (11), and the exciton component ${\mid {\Psi }_e\mid }^2$ is clearly seen to reflect these individual site energy fluctuations. The photon component ${\mid {\Psi }_p\mid }^2$, however, “responds” to spatially smoothed energetic disorder, which is illustrated in panel (a) with the diagonal disorder that is a result of box-averaging of the original site disorder with the box size of seven lattice sites.

Figure 4

Examples of the the time evolution of spatially identical wave packets built out of the polariton eigenstates of a perfect 1D microcavity as in Eq. (9) with the parameter ${\beta }^{1∕2}=5\mu \mathrm{m}$. For panels (a) and (b), the initial packet has zero total momentum, $k_0=0$; for panel (c) the initial packet has a finite momentum determined by $k_0={10}^4{\mathrm{cm}}^{-1}$. Only the photon part ${\mid {\Psi }_p\mid }^2$ of the polariton wave function is displayed. The initial packets are shown by long-dashed lines; results of the evolution after indicated times $t$ are shown by solid lines for the disordered system and by short-dashed lines for the perfect microcavity [except in panel (a), where the latter practically coincides with the initial packet].