Generation of entangled photon pairs from a single quantum dot embedded in a planar photonic-crystal cavity

We present a formal theory of single quantum dot coupling to a planar photonic crystal that supports quasidegenerate cavity modes and use this theory to describe and optimize entangled-photon pair generation via the biexciton-exciton cascade. In the generated photon pairs, either both photons are spontaneously emitted from the dot or one photon is emitted from the biexciton spontaneously while the other is emitted via the leaky cavity mode. In the strong-coupling regime, the generated photon pairs can be maximally entangled in qualitative agreement with the dressed-state predictions of Johne et al. [Phys. Rev. Lett. 100, 240404 (2008)]. We derive useful analytical formulas for the spectrum of the emitted photon pairs in the presence of exciton and biexciton broadening, which is necessary to connect to realistic experiments and demonstrate the important differences with the approximate dressed-state approach. We also present a method for calculating and optimizing the entanglement between the emitted photons, which can account for postsample spectral filtering. Pronounced entanglement values of greater than 80% are demonstrated using experimentally achievable parameters even without spectral filtering.

Figure 1

(Color online) (a) Schematic of the planar photonic crystal containing a single QD and (b) the resulting energy-level diagram for cavity-QED-assisted generation of entangled photons in the biexciton-exciton cascade decay. The biexciton state $|u⟩$ decays to the ground state $|g⟩$ via intermediate exciton state $|x⟩$ or $|y⟩$, emitting an $x$-polarized or $y$-polarized photon pair. The $x$-polarized and $y$-polarized cavity modes are coupled with the $|x⟩\to |g⟩$ and $|y⟩\to |g⟩$ transition, respectively. The vertical decays represent the leaky cavity mode decay $\left(\kappa \right)$ and the spontaneous decay from the background radiation modes $\left({\gamma }_b\right)$ (above the photonic-crystal slab light line).

Figure 2

(Color online) The spectra, $S_c\left(\omega \right)$ and $S_r\left(\omega \right)$, of the generated photons in the biexciton-exciton cascade decay for ${\delta }_x=0.1\text{meV}$, ${\Delta }_{xx}=1.0\text{meV}$, ${\gamma }_1=2{\gamma }_2=0.004\text{meV}$, ${\gamma }_b=\pi {\left|{\Omega }_{uk}\right|}^2=\pi {\left|{\Omega }_{gl}\right|}^2=0.05\mu \text{eV}$ (Ref. 28), $\kappa =0.05\text{meV}$, $g=0.11\text{meV}$, and ${\omega }_0=\left({\omega }_x+{\omega }_y\right)/2$. In (a) and (b), one photon is emitted from the biexciton decay and the other is emitted via the leaky cavity mode; in (c) and (d), both photons are radiated from the biexciton and exciton states via spontaneous (radiation-mode) decay. The other parameters are as follows: for (a) and (c), ${\Delta }_c^x=-{\Delta }_c^y={\delta }_x$ and for (b) and (d), ${\Delta }_c^x=-{\Delta }_c^y=-0.175\text{meV}$. The $x$-polarized photons are shown in blue and the $y$ polarized are shown in red; also, the solid (right) curves are for photons generated in the exciton decay and the dotted curves are for photons generated in the biexciton decay.

Figure 3

(Color online) The amplitude of the off-diagonal element of the density matrix for filtered photon pairs where ${\delta }_x$ is fixed at 0.1 meV. In (a) and (b) is shown the unfiltered and filter cases, respectively. The black curve represents ${\Delta }_c^x-{\Delta }_c^y=2{\delta }_x$ (case 1) and the red curve represents ${\Delta }_c^x-{\Delta }_c^y=-0.35\text{meV}$ (case 2); the other parameters are the same as in Fig. 2. The filter function corresponds to two spectral windows of width $w=0.2\text{meV}$ centered at ${\omega }_x^{-}$ and ${\omega }_u-{\omega }_x^{-}$. Note that ${\Delta }_c^x+{\Delta }_c^y=0$ corresponds to the optimal conditions for generating entangled photon pairs as shown in Fig. 2.

Figure 4

(Color online) [(a) and (b)] As in Figs. 2a, 2b but with $g=\kappa ={\delta }_x=0.05\text{meV}$. [(c) and (d)] As in Figs. 3a, 3b but with $g=\kappa ={\delta }_x=0.05$. The black curve represents ${\Delta }_c^x-{\Delta }_c^y=2{\delta }_x$ (case 1) and the red curve represents ${\Delta }_c^x-{\Delta }_c^y=-0.1\text{meV}$ (case 2). The filter function corresponds to two spectral windows of width $w=0.1\text{meV}$, centered at ${\omega }_x^{-}$ and ${\omega }_u-{\omega }_x^{-}$.

Figure 5

(Color online) [(a) and (b)] Same as in Fig. 2b but with $g_x=0.11\text{meV}$ and $g_y=0.08\text{meV}$ for (a) ${\Delta }_c^x=-{\Delta }_c^y=-0.122\text{meV}$, and for (b) ${\Delta }_c^x=-0.175\text{meV}$ and ${\Delta }_c^y=0.07\text{meV}$. [(c) and (d)] The values of $\left|\gamma \right|$ for generated photons by using two possible tuning methods, in (c) by changing ${\Delta }_c^x+{\Delta }_c^y$ for ${\Delta }_c^x-{\Delta }_c^y=0.245\text{meV}$ and in (d) by changing ${\Delta }_c^y$ for ${\Delta }_c^x=-0.175\text{meV}$. The chain curves represent results for filtered photons and the solid curves represent results for unfiltered photons; the filter function corresponds to two spectral windows of width $w=0.2\text{meV}$, centered at ${\omega }_x^{-}$ and ${\omega }_u-{\omega }_x^{-}$.