Entanglement dynamics and Mollow nonuplets between two coupled quantum dots in a nanowire photonic-crystal system

We introduce a nanowire-based photonic-crystal waveguide system capable of controllably mediating the photon coupling between two quantum dots which are macroscopically separated. Using a rigorous Green-function-based master equation approach, our two-dot system is shown to provide a wide range of interesting quantum transport regimes. In particular, we demonstrate the formation of long-lived entangled states and study the resonance fluorescence spectrum which contains clear signatures of the coupled quantum dot pair. Depending upon the operating frequency, one can obtain a modified Mollow triplet spectrum or a Mollow nonuplet, namely, a spectrum with nine spectral peaks. These multiple peaks are explained in the context of photon-exchange-mediated dressed states. Results are robust with respect to scattering loss, and spatial filtering via propagation allows for each quantum dot's emission to be observed individually.

Figure 1

(a) Proposed NW waveguide, which confines light to the central channel of reduced-radius NWs with the inset showing one of two QDs (size exaggerated) embedded in a NW and coupled with the other QD via the waveguide. (b) Energy-level diagram (not to scale) of a two-QD system interacting with the waveguide, with no drive and ${\delta }_{1,2}<0$, where $|{\psi }_{\pm }\rangle =\frac{1}{\sqrt{2}}\left(\right|0,1\rangle \pm |1,0\rangle )$.

Figure 2

(a) $|{\mathbf{E}}_{{\lambda }_1}{|}^2$ the in $y=0$ plane of the NW waveguide. QD locations are indicated via red circles. (b) $\mathrm{Im}\left\{G({\mathbf{r}}_{1/2},{\mathbf{r}}_{1/2};\omega )\right\}$, (c) $\mathrm{Im}\left\{G({\mathbf{r}}_1,{\mathbf{r}}_2;\omega )\right\}$, and (d) $-\mathrm{Re}\left\{G({\mathbf{r}}_1,{\mathbf{r}}_2;\omega )\right\}$, directly proportional to ${\mathrm{\Gamma }}_{1,1},{\mathrm{\Gamma }}_{1,2}$, and ${\delta }_{1,2}$. All rates are in units of $\mathrm{Im}\left\{G^{\mathrm{h}}(\mathbf{r},\mathbf{r};\omega )\right\}$. The crosses and circles indicate values at ${\lambda }_1$ and ${\lambda }_2$, respectively.

Figure 3

(a) Coupling dynamics between two QDs with QD 1 initially excited. The population of QD 1 (2) is shown in dark blue (dashed red), and entanglement is shown in light green. The dashed-dotted black line displays the population of QD 1 in the same structure without QD 2 for comparison. (b) Dynamics of an initially entangled pair state. Single QD population of the state initialized in $|{\psi }_{+}\rangle \left(\right|{\psi }_{-}\rangle )$ in dark blue (dashed-dotted black), and concurrence of $|{\psi }_{+}\rangle$ shown in light green.

Figure 4

(a) Population and concurrence when QD 1 is driven, following the labeling convention of Fig. 3. (b) Detected spectrum from QD 1 (2) in solid blue (dashed red). The QD 1 spectrum of an identical system without QD 2 is shown in dashed-dotted black. A Mollow triplet is only observed for the two-QD system under this excitation condition.

Figure 5

(a) Energy levels and expected transitions for a system evolving under $H_{\mathrm{eff}}$. Unprimed transitions are from the interaction picture, and primed are found when one considers the full Hamiltonian. The transitions between identical levels at ${\omega }_L$ are not labeled. (b) The detected spectrum from QD 1 (2) in solid blue (dashed red). The populations and concurrence are shown in the inset and again follow the convention of Fig. 3. The $a$ and $d$ transitions do not appear in the QD 2 spectrum as they have no effect on the state of QD 2. $\left|\mathbf{G}\right({\mathbf{r}}_D,{\mathbf{r}}_n;\omega \left)\right|$ increases with $\omega$ near ${\omega }_L$, causing spectral peak amplitudes to increase as well.