Light-mass Bragg cavity polaritons in planar quantum dot lattices

The exciton-polariton modes of a quantum dot lattice embedded in a planar optical cavity are theoretically investigated. Umklapp terms, in which an exciton interacts with many cavity modes differing by reciprocal lattice vectors, appear in the Hamiltonian due to the periodicity of the dot lattice. We focus on Bragg polariton modes obtained by tuning the exciton and the cavity modes into resonance at high symmetry points of the Brillouin zone. Depending on the microcavity design, these polariton modes at finite in-plane momentum can be guided and can have long lifetimes. Moreover, their effective mass can be extremely small, of the order of ${10}^{-8}{m}_0$ ($m_0$ is the bare electron mass), and they constitute the lightest excitonlike quasiparticles in solids.

Figure 1

Scheme of the system studied. An ideal quantum dot lattice is embedded in a planar microcavity of length $L_c$ and refractive index $n_R$.

Figure 2

(Color online) Scheme of photonic branches (solid lines) in a repeated zone scheme. The exciton energy band is indicated by the flat dashed line. Bragg polaritons are obtained by tuning the exciton energy at the zone boundary where photonic branches cross. Upper, lower, and central polariton modes (indicated by UP, LP, and CP) appear due to the light-matter coupling.

Figure 3

Reciprocal lattice of a square lattice (a) and of a hexagonal lattice with lattice constant $a$ (b). High symmetry points discussed in the text are indicated.

Figure 4

Bragg resonances at the $M$ point: $M_1$ labels the energy and momentum of Bragg polaritons obtained with the four nearest neighbor reciprocal lattice points (black dots). $M_2$ is a higher order Bragg resonance involving eight second nearest neighbor reciprocal lattice points (crosses).

Figure 5

Contour plot of the upper polariton dispersion at the $X$ point (a), $M$ point [$M_1$ in (b) and $M_2$ in (c)], and $W$ point (d). Note the mass anisotropy in the $X$-point case.

Figure 6

The energy dispersion of the Bragg polaritons at selected high symmetry points. $X$ polaritons along $\overline{\Gamma X}$ (a) and along $\overline{XM}$ (b), $M_1$ polaritons along $\overline{\Gamma M}$ (c) and the $q_x=q_y$ axis (d), and $W$ polaritons along the $q_x$ axis (e) and the $q_y$ axis (f). The dashed lines in (a) and (b) represent the uncoupled exciton and photon modes.

Figure 7

The exciton component of the lower (a), central (flat) (b), and upper (c) Bragg polaritons at the $X$ point.