Large Cross-Phase Modulations at the Few-Photon Level

We demonstrate an efficient cross-phase modulation (XPM) based on a closed-loop double-$\mathrm{\Lambda }$ system. The property of the double-$\mathrm{\Lambda }$ medium can be controlled by changing the phases of the applied optical fields. This phase-dependent XPM scheme can achieve large phase modulations at low-light intensities without requiring cavities or tightly focusing laser beams. With this scheme, we observe a $\pi$-level phase shift with two pulses, both consisting of eight photons in cold rubidium atoms. Such a novel scheme provides a simple route to generate strong interactions between photons and may have potential applications in all-optical quantum signal processing.

Figure 1

Energy level scheme and experimental apparatus. (a) Energy levels of the ${}^{87}\mathrm{Rb}$ $D_2$-line transition for the double-$\mathrm{\Lambda }$ experiment. Signal detuning $\mathrm{\Delta }$ is defined as ${\omega }_s-{\omega }_{24}$, where ${\omega }_s$ and ${\omega }_{24}$ are the frequencies of the signal field and the $|2⟩\leftrightarrow |4⟩$ transition, respectively. (b) Schematic diagram of the experimental setup. DL, diode laser; $\lambda /4$, quarter wave plate; $\lambda /2$, half wave plate; PL, polarizer; ND, neutral density filter; L, lens.

Figure 2

Observed EIT transmission versus probe field detuning. The black circles and red line represent the measurement data and theoretical curve, respectively. The inset shows the EIT transmission window. Probe detuning is defined as ${\omega }_p-{\omega }_{13}$, where ${\omega }_p$ and ${\omega }_{13}$ are the frequencies of the probe field and the $|1⟩\leftrightarrow |3⟩$ transition, respectively. The parameters for the theoretical curve are ${\alpha }_p=52$, $|{\mathrm{\Omega }}_c|=0.7\mathrm{\Gamma }$, ${\gamma }_{21}=0.001\mathrm{\Gamma }$, and ${\gamma }_{31}={\gamma }_{41}=1.25\mathrm{\Gamma }$.

Figure 3

Phase-dependent double-$\mathrm{\Lambda }$ experiment in the pulsed regime. The solid and dashed lines represent the experimental data and theoretical curves, respectively. The black (green) lines are the input probe (signal) pulses; the blue (red) lines are the transmitted probe (signal) pulses. The parameters for the theoretical curves (dashed lines) are ${\alpha }_p=46$, $|{\mathrm{\Omega }}_c|=|{\mathrm{\Omega }}_d|=0.7\mathrm{\Gamma }$, $\mathrm{\Delta }=13\mathrm{\Gamma }$, ${\gamma }_{21}=0.001\mathrm{\Gamma }$, and ${\gamma }_{31}={\gamma }_{41}=1.25\mathrm{\Gamma }$. (a) The relative phase ${\varphi }_r=1.5\mathrm{rad}$. (b) ${\varphi }_r=4.5\mathrm{rad}$. The peak powers of both the probe and signal pulses are around 1 nW, corresponding to around 40 000 photons per pulse.

Figure 4

Phase modulations at the few-photon level. Dependence of the transmission and the phase shift of the probe pulse on the relative phase ${\varphi }_r$. The numbers of both the probe and the signal photons are around 400 in (a) and (b), 40 in (c) and (d), and 8 in (e) and (f). Circles (squares) represent the transmission (phase shift) of the probe pulse. The black lines are the theoretical curves of the probe transmission and phase shift. The parameters for the theoretical curves are ${\alpha }_p=50$, $|{\mathrm{\Omega }}_c|=|{\mathrm{\Omega }}_d|=0.7\mathrm{\Gamma }$, $\mathrm{\Delta }=13\mathrm{\Gamma }$, ${\gamma }_{21}=0.001\mathrm{\Gamma }$, and ${\gamma }_{31}={\gamma }_{41}=1.25\mathrm{\Gamma }$.