Two-photon nonlinearity by an excitonic system placed in a cavity

We have investigated the two-photon nonlinearity attained by a finite-sized excitonic system placed inside of a cavity, paying attention to the dependence on the system size $N$. The two-photon nonlinearity is evaluated by utilizing semiclassical optical responses. In the quantum-dot limit, the excitonic system behaves as a single two-level system with an enhanced coupling to the cavity mode. In the bulk limit, the nonlinear phase shift decreases as $N$ is increased, proportionally to $N^{-1∕2}$.

Figure 1

Illustration of the exciton-cavity system. The cavity mode is connected through the right mirror to the external photon field. $g$, $\kappa$, and $V$ represent the coupling between the cavity mode and a single site, the escape rate of a cavity-mode photon, and the hopping interaction between sites, respectively.

Figure 2

The energy diagram for one-exciton eigenstates for the models (i), (ii), and (iii). The eigenenergy of the lowest one-exciton state is denoted by $\Lambda$. The higher-energy level of the one-exciton states in the model (iii) has $(N-1)$-fold degeneracy. The dashed lines connecting the cavity mode and the one-exciton states represent nonvanishing coupling.

Figure 3

Plot of $\mid \beta -1\mid$ as a function of photonic energy $q$. The photon energy is tuned to the lower Rabi-split frequency $(q\sim -\tilde{g})$ in (a) and to the higher one $(q\sim \tilde{g})$ in (b). Solid, dotted, and dashed lines are used to represent the results for the model (iii), (ii), and (i), respectively. The employed parameters are as follows: $N=5$, $\kappa ∕g=0.5$, $V∕g=5$; $\Omega$ is chosen so that the lowest excitonic energy becomes zero.

Figure 4

Plot of $\mid \beta -1\mid$ as a function of $q$, around the lower Rabi-split frequency. The coherent length of photons is chosen at $d=4∕\kappa$. The site number $N$ is 1 (solid line), 5 (dotted line), 10 (dashed line), and 20 (dot-dashed line). The system parameters are chosen as follows: $\kappa ∕g=0.5$, $\Omega =V$ (which results in $\Lambda =0$), and $V∕g=5$ in (a) and $V∕g=0$ in (b).

Figure 5

Plot of $\beta$ on the complex plane. The dotted line shows the unit circle. The integers near each point represent the system size $N$. For each $N$, the almost optimum pulse shape $(d=4∕\kappa ,q=-\tilde{g})$ is employed. The system parameters are the same as Fig. 4.

Figure 6

Plot of the nonlinear phase shift $\theta$ as a function of the system size $N$. The solid and dashed lines represent the results for $V∕g=5$ and 0, respectively. (Other parameters are the same as Fig. 5.) The thin dotted line is drawn by $\theta ={160}^{°}\times N^{-1∕2}$, by which numerical results are well fitted for large $N$.