Entanglement and lasing with two quantum dots in a microcavity

Entanglement and lasing of two quantum dots in a microcavity are investigated in the steady state achieved under a continuous incoherent pumping. Lasing is favored in the case of independent pumping of the dots, i.e., when an electron-hole pair which reaches one dot does not affect the other. In this case, having two dots allows a qualitative enhancement of the lasing emission as compared to the single dot case. On the other hand, when the two dots share the excitation bath, giving rise to a quantum uncertainty of the final state of the excitation, entanglement properties are enhanced in detriment of the photon emission. Slightly different couplings between the dots and the cavity induce good singlet-state population. This allows good entanglement in a system pumped continuously and incoherently.

Figure 1

Levels of the system up to two excitations in Dicke basis. Only the case with common pumping bath $(P_{\mathrm{ind}}=0)$ is presented here in order to illustrate the discussion in Sec 3. Solid lines represent the common pump, which only affects the triplet subspace $\mid T_{-1},n⟩$, $\mid T_0,n⟩$, and $\mid T_1,n⟩$ with $2P_{\mathrm{com}}$, increasing the dot excitations without changing the number of photons $n$. Dotted lines stand for the cavity photon decay. Dashed lines take account of the leaky modes affecting all QD levels. Finally, curved arrows show coherent couplings ($g$ and $\delta g$) between levels.

Figure 2

Scheme of the two QDs with their associated collection areas $A$, which can be deformed by applying an electric field (Ref. 13). The overlapping area is called $A_c$. When it is nonzero and comparable to the coherence length of the excitation, cross-terms of the pump operators appear in the master equation.

Figure 3

(Color online) Mean number of photons $⟨n⟩$ stored in the cavity as a function of the coupling of the second dot $g_2$ for $\kappa =1$, ${\Delta }_1={\Delta }_2=0$, and $P=0.33$ (all in units $g_1$). Both the cases of independent pumping only (red line, corresponding to $P_{\mathrm{ind}}=P$ and $P_{\mathrm{com}}=0$) and common pumping only (black and blue lines, with $P_{\mathrm{ind}}=0$ and $P_{\mathrm{com}}=P$) are plotted. In the latter case, the black line corresponds to $\gamma =0$ and the blue one to $\gamma =5\times {10}^{-3}$. The reference value of one QD in the cavity with $g=1$ is given. Inset: Population of the singlet state. In both plots, the black line has a discontinuity at $g_1=g_2$. The singular value assumed by $⟨n⟩$ and the singlet population in this case is marked by the black point.

Figure 4

(Color online) Tangle $\tau$ for various detunings as a function of $P_{\mathrm{com}}$. Parameters: $\kappa =1$, $\gamma =5\times {10}^{-3}$, $P_{\mathrm{ind}}=0$, and $g_2=0.6$ (all in units of $g_1$). The maximum tangle achieved grows with the detuning, but requires a larger pumping.

Figure 5

(Color online) Density plots of (a) mean number of photons, (b) population of the state $\mid S,0⟩$, (c) tangle, and (d) entropy, all as a function of $g_2$ and $P_{\mathrm{com}}$. Parameters are $\kappa =1$, $\gamma =5\times {10}^{-3}$, ${\Delta }_1={\Delta }_2=2$, and $P_{\mathrm{ind}}=0$ (all in units of $g_1$). The maximum value for the tangle $(\tau =0.64)$ is achieved at $g_2=0.6$ and $P_{\mathrm{com}}=1.22$ (this point is marked with a cross).

Figure 6

(Color online) Tangle and mean number of photons as a function of $g_2$ for $\kappa =1$, ${\Delta }_1={\Delta }_2=2$, $\gamma =5\times {10}^{-3}$, and total pump of 1.22 (all in units of $g_1$). The cases from independent pump $C=0$ (red) to common pump $C=1$ (dark blue) are considered. The intermediate curves correspond to $C=0.33$ (yellow), 0.66 (light blue), 0.82 (green), 0.91 (magenta), and 0.99 (black).

Figure 7

(Color online) Mean number of photons stored in the cavity as a function of pumping $P$ for $\kappa =0.2$, $\gamma =5\times {10}^{-3}$, and $g_1=g_2=1$. Both cases, with independent ($P_{\mathrm{ind}}=P$ and $P_{\mathrm{com}}=0$, red) and common pumping ($P_{\mathrm{ind}}=0$ and $P_{\mathrm{com}}=P$, green), are presented for the resonant case $({\Delta }_1={\Delta }_2=0)$. These are compared with the emission of a single QD in resonance with coupling $g=1$ (blue line), and the sum of emissions of two independent dots in resonance with renormalized coupling constants $g=\sqrt{2}$ (black line).

Figure 8

(Color online) (a) Mean number of photons $⟨n⟩$ stored in the cavity and (b) second-order coherence function $g^{\left(2\right)}\left(0\right)$ of the cavity field, as a function of $P_{\mathrm{ind}}=P$, for $P_{\mathrm{com}}=0$, $\kappa =0.2$, $\gamma =5\times {10}^{-3}$, and $g_2=1$. Several detuning cases are presented, with equal $({\Delta }_1={\Delta }_2=5)$, opposite $({\Delta }_1=-{\Delta }_2=5)$, and mixed (${\Delta }_1=0$, ${\Delta }_2=5$) configurations. There is a qualitative change with two dots as, even when none is in resonance with the cavity mode, the threshold for lasing (with linear increase of $⟨n⟩$ with $P$) is low also in the case where the single dot alone would not lase.