Tunable photon statistics in weakly nonlinear photonic molecules

In recent studies [Liew et al., Phys. Rev. Lett. 104, 183601 (2010); Bamba et al., Phys. Rev. A 83, 021802(R) (2011)], due to destructive interference between different paths for two-photon excitation, strong photon antibunching can be obtained in a photonic molecule consisting of two coupled cavity modes with weak Kerr nonlinearity when one of the cavity modes is driven resonantly. Here, we study the photon statistics in a nonlinear photonic molecule with both the two cavity modes being driven coherently. We show that the statistical properties of the photons can be controlled by regulating the strength ratio and the relative phase between the two driving fields. The photonic molecules with two driven modes can be used to generate tunable single-photon sources or controlled photonic quantum gates with weak Kerr nonlinearity.

Figure 1

Logarithmic plot (of base 10) of the equal-time second-order correlation function $g_a^{\left(2\right)}\left(0\right)$ as a function of the strength ratio between the driving fields $\eta$ and the detuning $\Delta /\kappa$ with the nonlinear interaction strength $U$ satisfying Eq. (25) in (a) or satisfying Eq. (3) in (b). The dash line in (a) [(b)] corresponds to the detuning $\Delta /\kappa$ satisfying Eq. (24) [Eq. (2)]. The parameters are $J=10\kappa ,{\varepsilon }_a=0.01\kappa$ and $\varphi =0$.

Figure 2

Logarithmic plot (of base 10) of the equal-time second-order correlation function $g_a^{\left(2\right)}\left(0\right)$ as a function of the strength ratio between the driving fields $\eta$ and the nonlinear interaction strength $U/\kappa$ with the detuning $\Delta /\kappa$ satisfying Eq. (24) in (a) or satisfying Eq. (2) in (b). The dash line in (a) [(b)] corresponds to the nonlinear interaction strength $U/\kappa$ satisfying Eq. (25) [Eq. (3)]. The parameters are the same as in Fig. 1.

Figure 3

(a) The optimal values of the detuning ${\Delta }_{\mathrm{opt}}/\kappa$ for the strong antibunching effect as functions of the strength ratio between the driving fields $\eta$ according to the numerical results (black solid line), Eq. (24) (red dash line), and Eq. (2) (blue short-dashed line). (b) The optimal values of the nonlinear interaction strength $U_{\mathrm{opt}}/\kappa$ for strong antibunching effect as functions of $\eta$ according to the numerical results (black solid line), Eq. (25) (red dashed line), and Eq. (3) (blue short-dashed line). The parameters are the same as in Fig. 1.

Figure 4

Logarithmic plot (of base 10) of the equal-time second-order correlation function $g_a^{\left(2\right)}\left(0\right)$ as a function of the relative phase $\varphi$ and the strength ratio ${\eta }^{-1}$ between the driving fields with the parameters $\Delta /\kappa$ and $U/\kappa$ given by Eqs. (24) and (25) in (a) or the parameters $\Delta /\kappa$ and $U/\kappa$ given by Eqs. (2) and (3) in (b). $g_a^{\left(2\right)}\left(0\right)$ as functions of the relative phase $\varphi$ for different values of the strength ratio $\eta$ is shown in (c) and (d), i.e., a few cuts taken from (a) and (b), respectively. The other parameters are $J=10\kappa$ and ${\varepsilon }_a=0.01\kappa$.

Figure 5

Logarithmic plot (of base 10) of (a) the equal-time second-order correlation function $g_a^{\left(2\right)}\left(0\right)$ and (b) the mean photon number $\langle n_a\rangle$ as functions of the strength ratio ${\eta }^{-1}$ for different values of the coupling constant $J/\kappa$ with the parameters $\Delta /\kappa$ and $U/\kappa$ given by Eqs. (2) and (3), ${\varepsilon }_a=0.01\kappa$ and the relative phase $\varphi =\pi /3$.

Figure 6

The equal-time second-order correlation function $g_a^{\left(2\right)}\left(0\right)$ plotted as a function of $({\Delta }_a-{\Delta }_{\mathrm{opt}})/\kappa$ for ${\Delta }_b={\Delta }_{\mathrm{opt}}$ and ${\kappa }_a={\kappa }_b=\kappa$ in (a), as a function of $({\Delta }_b-{\Delta }_{\mathrm{opt}})/\kappa$ for ${\Delta }_a={\Delta }_{\mathrm{opt}}$ and ${\kappa }_a={\kappa }_b=\kappa$ in (b), as a function of ${\kappa }_a/\kappa$ for ${\Delta }_a={\Delta }_b={\Delta }_{\mathrm{opt}}$ and ${\kappa }_b=\kappa$ in (c), and as a function of ${\kappa }_b/\kappa$ for ${\Delta }_a={\Delta }_b={\Delta }_{\mathrm{opt}}$ and ${\kappa }_a=\kappa$ in (d). The other parameters are $U=U_{\mathrm{opt}},J=10\kappa ,{\varepsilon }_a=0.01\kappa ,\eta =2$ and $\varphi =0$. ${\Delta }_{\mathrm{opt}}$ and $U_{\mathrm{opt}}$ are given by Eqs. (24) and (25).

Figure 7

The second-order correlation function $g_a^{\left(2\right)}\left(\tau \right)$ plotted as a function of the normalized time delay $\tau /(2\pi /J)$ for the strength ratio $\eta$ and the relative phase $\varphi$ between the driving fields taking different values, (a) $\eta =2$ and $\varphi =0$, (b) $\eta =3$ and $\varphi =0$, (c) $\eta =2$ and $\varphi =0.1$, (d) $\eta =2$ and $\varphi =0.15$. The other parameters are $J=10\kappa ,{\varepsilon }_a=0.01\kappa ,U=U_{\mathrm{opt}}$ and $\Delta ={\Delta }_{\mathrm{opt}}$ with ${\Delta }_{\mathrm{opt}}$ and $U_{\mathrm{opt}}$ given by Eqs. (24) and (25).