Low-light-level four-wave mixing by quantum interference

We observed electromagnetically induced transparency-based four-wave mixing (FWM) in the pulsed regime at low light levels. The FWM conversion efficiency of $3.8\left(9\right)\%$ was observed in a four-level system of cold ${}^{87}\mathrm{Rb}$ atoms using a driving laser pulse with a peak intensity of $\approx 80$ $\mu \mathrm{W}/{\mathrm{cm}}^2$, corresponding to an energy of $\approx 60$ photons per atomic cross section. Comparison between the experimental data and the theoretical predictions proposed by Harris and Hau [Phys. Rev. Lett. 82, 4611 (1999)] showed good agreement. Additionally, a high conversion efficiency of $46\left(2\right)\%$ was demonstrated when applying this scheme using a driving laser intensity of $\approx 1.8$ $\mathrm{mW}/{\mathrm{cm}}^2$. According to our theoretical predictions, this FWM scheme can achieve a conversion efficiency of nearly $100\%$ when using a dense medium with an optical depth of 500.

Figure 1

Energy level scheme and experimental apparatus. (a) Energy levels of ${}^{87}\mathrm{Rb}$ $D_2$-line transition for the EIT-based FWM experiment. Driving detuning, $\Delta$, is defined as ${\omega }_d-{\omega }_{24}$, where ${\omega }_d$ and ${\omega }_{24}$ are the frequencies of the driving field and the $|2\rangle \leftrightarrow |4\rangle$ transition, respectively. (b) Schematic diagram of the experimental setup. $\lambda$/4, quarter-wave plate; POL, polarizer; PMT, photomultiplier tube; and PH, 200-$\mu$m pinhole for blocking the scattering light of the coupling and driving fields.

Figure 2

Four-wave mixing transmission of the probe and signal pulses as a function of the driving Rabi frequency (${\Omega }_d$). The blue squares and the red circles show the transmission of the probe and the signal pulses, respectively. The solid lines show the theoretical curves with the parameters $\Delta =13\Gamma$, $\alpha$ = 42, ${\Omega }_c=0.32\Gamma$, and ${\gamma }_{21}=9\times {10}^{-4}\Gamma$. Considering all data in this measurement, ${\gamma }_{21}$ has a standard deviation of $7\times {10}^{-4}\Gamma$; hence we also plot the theoretical curves (dashed lines) with the parameters ${\gamma }_{21}=1.6\times {10}^{-3}\Gamma$ (lower dashed lines) and ${\gamma }_{21}=2\times {10}^{-4}\Gamma$ (upper dashed lines). The error bars represent $\pm 1$ standard deviation based on measurement statistics.

Figure 3

Probe and signal pulses propagating through the EIT-based FWM medium. The gray (upper) lines are the incident probe pulses. The red (middle) and blue (lower) lines are the generated signal and the transmitted probe pulses, respectively. The dashed lines plot the theoretical curves with the parameters (a) $\Delta =13\Gamma$, $\alpha =40$, ${\Omega }_c=0.32\Gamma$, ${\Omega }_d=0.37\Gamma$, and ${\gamma }_{21}=1\times {10}^{-3}\Gamma$. (b) $\Delta =13\Gamma$, $\alpha =42$, ${\Omega }_c=0.32\Gamma$, ${\Omega }_d=0.67\Gamma$, and ${\gamma }_{21}=1.5\times {10}^{-3}\Gamma$. The FWM conversion efficiencies in (a) and (b) were 42$\%$ and 13$\%$, respectively.

Figure 4

Four-wave mixing conversion efficiency as a function of the driving detuning ($\Delta$). The blue squares and red circles show the transmission of the probe and signal pulses, respectively. The dashed blue (probe) and solid red (signal) lines are the theoretical curves with the parameters $\alpha =36$, ${\Omega }_c=0.32\Gamma$, ${\Omega }_d=0.35\Gamma$, and ${\gamma }_{21}=6\times {10}^{-4}\Gamma$. The error bars represent $\pm 1$ standard deviation based on measurement statistics.

Figure 5

Nearly $100\%$ FWM conversion efficiency with a great optical depth. The dashed blue (probe) and solid red (signal) lines are the theoretical curves plotted according to Eqs. (6) and (7) with the parameters $\alpha =500$ and ${\Omega }_c={\Omega }_d=0.32\Gamma$.