Interacting dark states with enhanced nonlinearity in an ideal four-level tripod atomic system

We present an experimental system to generate large cross-phase modulation (XPM) in cold rubidium atoms. By using an efficient state-preparation technique in the ${}^{87}\mathrm{Rb}$ $D1$ line, an ideal four-level tripod-type atomic system is formed, which generates large cross-Kerr nonlinearity via interacting dark states in this system. The induced phase shift due to XPM for the probe beam is measured for different trigger beam intensities, which is the key to achieving conditional quantum phase gates and many other applications in quantum information processing.

Figure 1

(Color online) (a) Relevant energy diagram of the $D1$ line in the ${}^{87}\mathrm{Rb}$ atom. Solid line: transition for the left circularly polarized probe laser beam $\left({\Omega }_P\right)$; dotted line: transition for the right circularly polarized trigger beam $\left({\Omega }_T\right)$; dashed-dot line: transition for the left circularly polarized coupling beam $\left({\Omega }_C\right)$; dashed line: transition for the right circularly polarized pumping beam $\left({\Omega }_{\text{pump}}\right)$. (b) Experimental arrangement of laser beams.

Figure 2

Measured probe transmissivity as a function of the probe frequency detuning. The frequency detunings are (a) ${\Delta }_C=5\mathrm{MHz}$, ${\Delta }_T=0\mathrm{MHz}$; (b) ${\Delta }_C=0\mathrm{MHz}$, ${\Delta }_T=-5\mathrm{MHz}$; (c) ${\Delta }_C=5\mathrm{MHz}$, ${\Delta }_T=-5\mathrm{MHz}$; (d) ${\Delta }_C={\Delta }_T=0\mathrm{MHz}$. The Rabi frequencies of coupling and trigger beams are ${\Omega }_T\approx {\Omega }_C\approx 7\mathrm{MHz}$.

Figure 3

(Color online) The probe transmission for (a) only with coupling beam, (b) only with trigger beam, and (c) with both coupling and trigger beams, respectively. The (red) dotted lines are the theoretically calculated results with the parameters ${\Omega }_C=6.5\mathrm{MHz}$, ${\Omega }_T=5.7\mathrm{MHz}$, and $P_0{N}_{F1}=0.28\times {10}^{11}∕{\mathrm{cm}}^3$ for (a) and (b), $P_0{N}_{F1}=0.20\times {10}^{11}∕{\mathrm{cm}}^3$ for (c). Other parameters are ${\gamma }_{ab}=1.8\mathrm{MHz}$ and ${\gamma }_{ac}=3.6\mathrm{MHz}$ (${\gamma }_{ab}$ and ${\gamma }_{ac}$ include the laser linewidths).

Figure 4

(Color online) Mach-Zehnder interferometer to detect the dispersion of the probe beam in the system.

Figure 5

(Color online) Measured dispersion curves (solid line) for without (a) and with (b) the trigger beam, (red) dotted lines are the theoretically calculated results with the parameters ${\Omega }_C=6.5\mathrm{MHz}$ and ${\Omega }_T=5.7\mathrm{MHz}$. Other parameters are the same as in Fig. 3.

Figure 6

Measured phase shifts as a function of the trigger beam intensity.