Entangled photon pairs from a quantum-dot cascade decay: The effect of time reordering

Coulomb interactions between confined carriers remove degeneracies in the excitation spectra of quantum dots. This provides a which-path information in the cascade decay of biexcitons, thus spoiling the energy-polarization entanglement of the emitted photon pairs. We theoretically analyze a strategy of color coincidence across generation (AG), recently proposed as an alternative to the previous within generation approach. We simulate the system dynamics and compute the correlation functions within the density-matrix formalism. This allows estimations of quantities that are accessible by a polarization-tomography experiment and that enter the expression of the two-photon concurrence. We identify the optimum parameters within the AG approach and the corresponding maximum values of the concurrence.

Figure 1

(Color online) Level structure of the quantum dot. The fine-structure splitting and the biexciton binding energy are given, respectively, by ${\delta }_{HV}=E_3-E_2$ and ${\Delta }_B=E_2+E_3-E_4$. In the WG strategy, the ideal case corresponds to the biexciton and exciton emission frequencies being independent on the polarization: ${\omega }_1\equiv {\omega }_p={\omega }_q$ and ${\omega }_2\equiv {\omega }_r={\omega }_s$ $\left({\delta }_{HV}=0\right)$. In the AG approach the photon emitted in the $B\to X_H$ $\left(B\to X_V\right)$ transition matches the color of that emitted by the $X_V\to G$ $\left(X_H\to G\right)$ decay $\left({\Delta }_B=0\right)$. Therefore, ${\omega }_1\equiv {\omega }_p={\omega }_s$ (cyan arrows) and ${\omega }_2\equiv {\omega }_q={\omega }_r$ (red arrows). In order to erase the which-path information provided by the emission order of the ${\omega }_1$ and ${\omega }_2$ photons, which is opposite in the two paths, the photons are delayed by ${\delta }_{\chi }$ $\left(\chi =p,q,r,s\right)$.

Figure 2

(Color online) Values of $\left|{\rho }_{HV}\right|={\mathcal{C}}^M\left(\mathbf{\Gamma }\right)/2$ as a function of ${{\rm Y}}_{sp}$ and ${{\rm Y}}_{qp}$ (where ${{\rm Y}}_{\chi {\chi }^{\prime }}={\Gamma }_{\chi }/{\Gamma }_{{\chi }^{\prime }}$). The value of $\text{log}\left({{\rm Y}}_{rp}\right)$ has been fixed to (a) $-1.2$, (b) 0, and (c) 1.2.

Figure 3

(Color online) The solid black lines give the values of the concurrence for optimized delays, ${\mathcal{C}}^M$, as a function of the ratio ${{\rm Y}}_{\chi ,{\chi }^{\prime }}={\Gamma }_{\chi }/{\Gamma }_{{\chi }^{\prime }}$ and for different constraints. (a) ${\Gamma }_s={\Gamma }_p$ and ${\Gamma }_r={\Gamma }_q$, with $\left(\chi ,{\chi }^{\prime }\right)=\left(q,p\right)$; (b) ${\Gamma }_s={\Gamma }_q$ and ${\Gamma }_r={\Gamma }_p$, with $\left(\chi ,{\chi }^{\prime }\right)=\left(p,q\right)$; (c) ${\Gamma }_q={\Gamma }_p$ and ${\Gamma }_s={\Gamma }_r$, with $\left(\chi ,{\chi }^{\prime }\right)=\left(p,r\right)$. The dashed lines are the values of the upper limits for the concurrence ${\mathcal{C}}_0$, corresponding to a complete cancellation of the which-path information. Photon emission rates that are set equal are denoted by arrows of equal thickness in the level schemes on the right-hand side of each plot.