Strongly entangled light from planar microcavities

The emission of entangled light from planar semiconductor microcavities is studied and the entanglement properties are analyzed and quantified. Phase matching of the intracavity scattering dynamics for multiple pump beams or pulses, together with the coupling to external radiation, leads to the emission of a manifold of entangled photon pairs. A decomposition of the emitted photons into two parties leads to a strong entanglement of the resulting bipartite system. For the quantification of the entanglement, the Schmidt number of the system is determined by the construction of Schmidt number witnesses. It is analyzed to which extent the resources of the originally strongly entangled light field are diminished by dephasing in propagation channels.

Figure 1

Sketch of the considered physical processes. The inset (a) shows the dispersion relations of the excitons, cavity photons, and polaritons. Part (b) visualizes the interbranch polariton pair scattering. Panel (c) depicts the emission and propagation of the emitted entangled light.

Figure 2

Coefficients of the Hopfield transformation matrix in Eq. (7), for the parameters $E_C\left(0\right)=1.5$ eV and ${\Omega }_R=2$ meV. The solid lines correspond to $M_{11\mathbf{k}}^2$ and the dashed ones to $M_{12\mathbf{k}}^2$. The color of the curves indicates the values of the normalized detuning $\delta$. For large values of $\left|\mathbf{k}\right|$ the coefficient $M_{11\mathbf{k}}^2$ converges to one and $M_{12\mathbf{k}}^2$ vanishes.

Figure 3

Magnitude of ${\beta }^2$ in the $(\delta ,p_s)$ plane according to Eq. (18b) for $E_C\left(0\right)=1.5$ eV, $E_b=10$ meV, and ${\mathbf{k}}_p=0.05k_0{\mathbf{e}}_x$. The phase-matching scattering wave vector $\mathbf{q}$ follows from Eqs. (19) and (21).

Figure 4

Amount of entanglement within the state $\overline{\rho }(t_1,t_2)$ depending on $\Delta t$ and quantized by the SN. We apply three different pumps with wave vectors ${\mathbf{k}}_{pn}=0.025n{k}_0{\mathbf{e}}_x$, $n=1,2,3$ to the cavity. The system parameters are $E_C\left(0\right)=1.5$ eV and $E_b=10$ meV. The dispersion of the medium is chosen to be $\omega \left(k\right)=0.5k$. Based on our units $\hslash =c=1$, we choose a typical reference time scale $t_0=1$ eV. The inset shows the behavior for weak dephasing.

Figure 5

Amount of entanglement within the state $\overline{\rho }(t_1,t_2)$ for $N=4$ different pumps. The parameters are the same as in Fig. 4, aside from the additional ${\mathbf{k}}_{pn}$ for $n=4$.