Excitons in cores of exciton-polariton vortices

The inner structure of vortices in Bose-Einstein condensates of exciton polaritons is studied theoretically. We show analytically that the healing lengths for the exciton and photon components of exciton-polariton condensate are essentially different. Namely, the exciton healing length may be about two orders of magnitude smaller than the photon healing length. Experimentally, in near-field photoluminescence, the photon part of the exciton-polariton condensate is detected. The suggested theory shows that the cores of experimentally observed vortices are photon cores, and there could be thousands of unobserved exciton polaritons with a strongly reduced photonic fraction inside them.

Figure 1

Ratio between the healing lengths for the exciton and photon components of the polariton condensate ${\xi }_{\mathrm{ex}}/{\xi }_{\mathrm{ph}}$ versus $a=2(E_0-\mu )/\hslash {\Omega }_R$. As $a$ increases, the ratio slowly grows yet always remains much smaller than unity, and steeply rises to infinity only when $a$ approaches its limit 1 (see inset), which is never physically realized (see the text).

Figure 2

Analytical solution of the set of equations (19) for $a=0.1$ (for this value, $\tilde{\eta }=5$ corresponds to $\approx$15 $\mu$m). The function $f_{\psi }$ (blue dotted line) is given by (20). The function $f_{\chi }$ (red solid line) is calculated as an exact solution of the cubic Eq. (17). Black dashed line represent the approximation for $f_{\chi }$ given by the formula (18) which appears to be invalid in the region of the exciton vortex core (grey field); therefore these dependencies can be considered correct only starting from the region of the photon vortex core characteristic size and further.

Figure 3

Vortical solutions of the set of GP-like equations for the two-component polariton condensate as functions of the radial coordinate measured in the units of the exciton condensate healing length $\eta =r/{\xi }_{\mathrm{ex}}$ for (a) $a=0.1$, (b) $a=0.5$, (c) $a=0.9$. Red circles represent $f_{\chi }\left(\eta \right)$; black squares represent $f_{\psi }\left(\eta \right)$.

Figure 4

(a) The (normalized) densities of the photon and exciton components of the condensate as functions of distance from the vortex center for $a=0.5$. The densities are given by $n_{\chi }\left(r\right)=n_{\chi 0}{f}_{\chi }^2$ and $n_{\psi }\left(r\right)=n_{\psi 0}{f}_{\psi }^2$, respectively, where $n_{\chi 0}$ and $n_{\psi 0}$ are the densities of the uniform condensates which ratio equals $a^2$ (see the text). The distance $r$ is calculated from the dimensionless value $\eta =r/{\xi }_{\mathrm{ex}}$ for parameter values $\hslash {\Omega }_R=5$ meV, $m_{\mathrm{ex}}=0.153m_e$. (b) The core region.