Highly efficient four-wave mixing in double-$\Lambda$ system in ultraslow propagation regime

We perform a time-dependent analysis of four-wave mixing (FWM) in a double-$\Lambda$ system in an ultraslow-propagation regime and obtain the analytical expressions of pulsed probe laser, FWM-generated pulse, phase shifts and absorption coefficients, group velocities, and FWM efficiency. With these analytical expressions, we show that an efficiently generated FWM field can acquire the same ultraslow group velocity $(V_g∕c\sim {10}^{-4}–{10}^{-5})$ and pulse shape of a probe pump and that the maximum FWM efficiency is greater than 25%, which is orders of magnitude larger than previous FWM schemes in the ultraslow-propagation regime.

Figure 1

Schematic of FWM in a double-$\Lambda$ system. Lifetime-broadened four-level atoms interact with two cw laser pump fields (frequencies ${\omega }_c$ and ${\omega }_d$ and Rabi frequencies $2{\Omega }_c$ and $2{\Omega }_d$) and a weak pulsed probe pump (frequency ${\omega }_p$ and Rabi frequency $2{\Omega }_p$) to generate a FWM-generated pulsed field (frequency ${\omega }_m$ and Rabi frequency $2{\Omega }_m$).

Figure 2

The ratio of absorption coefficients ${\alpha }_{-}∕{\alpha }_{+}$ versus $\mid {\Omega }_d\mid ∕{\gamma }_2$ for $\mid {\Omega }_c\mid ∕{\gamma }_2=2$ (uppermost), 1 (middle), and 0.4 (lowest). The inset shows the lowest curve $\mid {\Omega }_c\mid ∕{\gamma }_2=0.4$ for $\mid {\Omega }_b\mid ∕{\gamma }_2$ in the small interval [0.4, 0.401]. The other parameters are ${\kappa }_{02}={\kappa }_{03}$, ${\gamma }_1∕{\gamma }_2={10}^{-4}$, ${\gamma }_3∕{\gamma }_2=2.5∕1.2$, $\Delta {\omega }_c=\Delta {\omega }_p=0$, and $\Delta {\omega }_d=0.02{\gamma }_2$.

Figure 3

${\mid {\Lambda }_m(0,\omega )\mid }^2$ versus $\omega ∕{\gamma }_2$ with ${\Lambda }_p(0,\omega )={\Omega }_{p0}\tau \sqrt{\pi }\exp [-{\left(\omega \tau \right)}^2∕4]\iff \Omega (0,t)={\Omega }_{p0}\exp (-t^2∕{\tau }^2)$. The solid-line curve represents the exact solution (7b) while the dashed-line curve denotes the approximated solution by neglecting the $\exp \left(i{K}_{-}z\right)$ term and taking $U_{\pm }\simeq W_{\pm }$ and $K_{+}\simeq K_{+}\left(0\right)+K_{+}^{\left(1\right)}\omega$ in Eq. (7b). The parameters are $\tau ={10}^{-6}\mathrm{s}$ and ${\Omega }_c={\Omega }_b={\gamma }_2$.

Figure 4

The relative group velocity $V_g∕c$ versus $\mid {\Omega }_d\mid ∕{\gamma }_2$ with ${\Omega }_c=0.5{\gamma }_2$ (thick line) and ${\Omega }_c={\gamma }_2$ (thin line). The parameters ${\kappa }_{02}={\kappa }_{03}=2\times {10}^9{\mathrm{cm}}^{-1}{\mathrm{s}}^{-1}$, ${\gamma }_1∕{\gamma }_2={10}^{-4}$, ${\gamma }_3∕{\gamma }_2=2.5∕1.2$, $\Delta {\omega }_c=\Delta {\omega }_p=0$, $\Delta {\omega }_d=0.02{\gamma }_2$, and ${\gamma }_2\simeq 0.6\times {10}^8{\mathrm{s}}^{-1}$.

Figure 5

FWM efficiency $\eta$ versus $\mid {\Omega }_d\mid ∕{\gamma }_2$ for $\mid {\Omega }_c\mid ∕{\gamma }_2=2$ (thick solid line), 1 (dashed line), and 0.5 (thin solid line). The left (right) panel has ${\kappa }_{02}={\kappa }_{03}=2\times {10}^9{\mathrm{cm}}^{-1}{\mathrm{s}}^{-1}$ $({\kappa }_{03}=4{\kappa }_{02}=2\times {10}^9{\mathrm{cm}}^{-1}{\mathrm{s}}^{-1})$. The other parameters for both panels are ${\gamma }_1∕{\gamma }_2={10}^{-4}$, ${\gamma }_3∕{\gamma }_2=2.5∕1.2$, $\Delta {\omega }_c=\Delta {\omega }_p=0$, $\Delta {\omega }_d=0.02{\gamma }_2$, ${\gamma }_2\simeq 0.6\times {10}^8{\mathrm{s}}^{-1}$, ${\omega }_m=1.1{\omega }_p$ and $L=1\mathrm{cm}$.