Polarization entanglement visibility of photon pairs emitted by a quantum dot embedded in a microcavity

We study the photon emission from a quantum dot embedded in a microcavity. Incoherent pumping of its excitons and biexciton provokes the emission of leaky and cavity modes. By solving a master equation we obtain the correlation functions required to compute the spectrum and the relative efficiency among the emission of pairs and single photons. A quantum regime appears for low pumping and large rate of emission. By means of a post-selection process, a two beams experiment with different linear polarizations could be performed producing a large polarization entanglement visibility precisely in the quantum regime.

Figure 1

(Color online) Ladder of levels (black continuous lines) for a four-state QD coupled to a cavity mode with the two possible $(R,L)$ circular polarizations. States are labeled as described in the text with $G{n}_R{n}_L$ $(n_R,n_L=0,1,2,\dots )$. Double (purple) continuous lines depict the coupling $q_i,J$, dashed (red) lines the pumping with rate $P$, dotted (blue) lines the leaky modes emission with rate $\gamma$, and dashed-dotted (blue) lines the emission of cavity modes with rate $\kappa$. Pumping and emission lines in the upper part of the diagram are not plotted in order to simplify the figure.

Figure 2

Spectrum (in arbitrary units) of the emission of cavity photons for ${\omega }_C=1000$, $q=0.1$, ${\Delta }_1=-{\Delta }_2=0.5$, $\gamma =0.01$, $\kappa =0.1$ and $P=0.05$ and $P=10.0$ (dashed line) with all the magnitudes in meV. ${\omega }_X={\omega }_C+{\Delta }_1$ and ${\omega }_B={\omega }_C+{\Delta }_2$ label the frequencies for recombination of the exciton and biexciton, respectively (see text).

Figure 3

(Color online) Second order coherence functions $g_{R,R}^{\left(2\right)}$ and $g_{R,L}^{\left(2\right)}$ describing the photon pair emission efficiency $(\propto G^{\left(2\right)})$ compared with the single photon emission probability $(\propto G^{\left(1\right)})$. $q=0.1\mathrm{meV}$, ${\Delta }_1=-{\Delta }_2=0.5\mathrm{meV}$, and $\gamma =0.01\mathrm{meV}$. The line where $g_{R,R}^{\left(2\right)}=1$ is labeled.

Figure 4

(Color online) Second order coherence functions $g_{R,R}^{\left(2\right)}$ and $g_{R,L}^{\left(2\right)}$ describing the photon pair emission efficiency $(\propto G^{\left(2\right)})$ compared with the single photon emission probability $(\propto G^{\left(1\right)})$. ${\Delta }_1=-{\Delta }_2=5$, $\gamma =0.1$ in units of $q$. The line where $g_{R,R}^{\left(2\right)}=1$ is labeled.

Figure 5

(Color online) Polarization entanglement visibility $\mathcal{V}$ for a pair of photons with two different linear polarizations. $q=0.1\mathrm{meV}$, ${\Delta }_1=-{\Delta }_2=0.5\mathrm{meV}$, $\gamma =0.01\mathrm{meV}$.

Figure 6

(Color online) Polarization entanglement visibility $\mathcal{V}$ for a pair of photons with two different linear polarizations. ${\Delta }_1=-{\Delta }_2=5$, $\gamma =0.1$ in units of $q$.