Plasmon-assisted two-photon Rabi oscillations in a semiconductor quantum dot – metal nanoparticle heterodimer

Two-photon Rabi oscillations hold potential for quantum computing and quantum information processing because during a Rabi cycle a pair of entangled photons may be created. We theoretically investigate the onset of this phenomenon in a heterodimer comprising a semiconductor quantum dot strongly coupled to a metal nanoparticle. Two-photon Rabi oscillations in this system occur due to a coherent two-photon process involving the ground-to-biexciton transition in the quantum dot. The presence of a metal nanoparticle near the quantum dot results in a self-action of the quantum dot via the metal nanoparticle because the polarization state of the latter depends on the quantum state of the former. The interparticle interaction gives rise to two principal effects: (i) enhancement of the external field amplitude and (ii) renormalization of the quantum dot's resonance frequencies and relaxation rates of the off-diagonal density matrix elements, both depending on the populations of the quantum dot's levels. Here, we focus on the first effect, which results in interesting features, in particular, an increased number of Rabi cycles per pulse compared to an isolated quantum dot and subsequent growth of the number of entangled photon pairs per pulse. We also discuss the destructive role of radiative decay of the excitonic states on two-photon Rabi oscillations for both an isolated quantum dot and a heterodimer.

Figure 1

(a) Schematics of a SQD-MNP heterodimer subject to a pulsed applied field ${\mathcal{E}}_0\left(t\right)={\mathit{E}}_0\left(t\right)cos\left({\omega }_{0}t\right)$. The field is linearly polarized along the system axis (shown by the red arrow). Here, $d$ is the SQD-MNP center-to-center distance, $r$ is the radius of the MNP, ${\varepsilon }_s$ and ${\varepsilon }_m\left(\omega \right)$ are the dielectric constants of the SQD and the MNP, respectively. The system is embedded in an isotropic and nonabsorbing medium with permittivity ${\varepsilon }_b$. (b) Energy diagrams of the MNP (left) and a ladder-type three-level SQD (right). The excited state of the MNP represents a broad line centered at the frequency of the LSP's resonance ${\omega }_{\mathrm{LSP}}$, shown by the dashed yellow line. For the SQD, $|1\rangle ,|2\rangle$, and $|3\rangle$ are the ground, one-exciton, and biexciton states, respectively. The energies of these states are ${\varepsilon }_1=0,{\varepsilon }_2=\hslash {\omega }_2$, and ${\varepsilon }_3=2\hslash ({\omega }_2-{\mathrm{\Delta }}_B/2)$, where $\hslash {\mathrm{\Delta }}_B$ is the biexciton binding energy. Allowed transitions with the corresponding transition dipole moments ${\mathbf{\mu }}_{21}$ and ${\mathbf{\mu }}_{32}$ are indicated by solid double-headed arrows. Wiggly arrows denote the spontaneous decay with rates ${\gamma }_{32}$ and ${\gamma }_{21}$. The black dashed line shows the location of the coherent two-photon resonance ${\omega }_3/2={\omega }_2-{\mathrm{\Delta }}_B/2$ (with simultaneous absorption of two photons).

Figure 2

Contour plots of the TPRO of an isolated SQD. Top: population of the biexciton state ${\rho }_{33}$. The white dashed curve shows the $t_0$ dependence of the incident pulse area $A$, at which ${\rho }_{33}$ acquires its first maximum, plotted according to Eq. (14) with a correction coefficient of 0.62. Bottom: population of the one-exciton state ${\rho }_{22}$.

Figure 3

Area dependence of the populations of the bi- and one-exciton states for the different incident pulse durations indicated above each panel. ${\mathrm{\Delta }}_B=30{\mathrm{ps}}^{-1}$.

Figure 4

Population dynamics calculated for the incident pulse with area $A=9\pi$, duration $t_0=20/{\mathrm{\Delta }}_B=2/3$ ps, and delay time $t_d=2$ ps. The red dotted curve shows the profile of the incident Gaussian pulse, $exp\{-{[(t-t_0)/t_d]}^2\}$.

Figure 5

Same as in Fig. 4, but for $A=50\pi ,t_0=900/{\mathrm{\Delta }}_B=30$ ps, and $t_d=100$ ps.

Figure 6

Contour plots of the TPRO of a SQD-MNP hybrid calculated for a center-to-center distance $d=18$ nm. Top: population of the biexciton state ${\rho }_{33}$. The dashed white curve shows the $t_0$ dependence of the incident pulse area $\tilde{A}$, at which ${\rho }_{33}$ acquires its first maximum, plotted according to Eq. (15) with a correction prefactor of 0.62. Bottom: population of the one-exciton state ${\rho }_{22}$.

Figure 7

Same as in Fig. 3, but for a hybrid with center-to-center distance $d=18$ nm.

Figure 8

Same as in Fig. 4, but for a hybrid with center-to-center distance $d=18$ nm.

Figure 9

Same as in Fig. 5, but for a hybrid with center-to-center distance $d=18$ nm.

Figure 10

TPRO in the SQD-MNP hybrid considered in this paper as a function of the interparticle distance $d$ and the pulse area $A$. Plotted is the biexciton state population ${\rho }_{33}$ for a hybrid calculated numerically for an incident pulse with $t_0=20/{\mathrm{\Delta }}_B=(2/3)$ ps.