Biphoton generation in a two-level atomic ensemble

We have theoretically studied the space-time entangled biphoton state generated from a two-level atomic system. In the photon counting measurement, the two-photon coincidence counting rate is a damped oscillation. The oscillation period is determined by the effective Rabi frequency and the damping rate is determined by the linewidth of the inhomogeneous-broadened ground state and the dipole dephasing rate. In an optical-pathway-balanced configuration, the two-photon temporal correlation shows an antibunching effect which corresponds to the interference between two types of nonlinear four-wave mixing processes occurring in such a two-level system. The visibility of the normalized second-order quantum coherence function $g^{\left(2\right)}\left(\tau \right)$ increases along with the increase of the effective Rabi frequency, but has an upper limit at 45%. We find agreement between the theory and the experiments [P. Kolchin et al., Phys. Rev. Lett. 97, 113602 (2006); S. Du, J.-M. Wen, M. H. Rubin, and G. Y. Yin, ibid 98, 053601 (2007)].

Figure 1

(Color online) (a) Four-wave mixing in a two-level system. In the presence of a retroreflected pump beam at frequency ${\omega }_1$, the two-level atoms scatter light into correlated fields at the frequencies ${\omega }_1+\delta$ and ${\omega }_1-\delta$. The pump detuning is $\Delta ={\omega }_1-{\omega }_{eg}$. (b) In the photon counting measurement, detectors D1 and D2 are applied to collect paired photons propagating in opposite directions.

Figure 2

(Color online) (a) Resonance around $\delta =0$. In such a case, the generated beams are scattered at the same frequency ${\omega }_1$ as the pump laser. (b) Resonances around $\delta =\pm {\Omega }_e$. In such a case, the pump laser creates two virtual states, shown as dashed lines, one at an energy $\hslash {\omega }_1$ above the ground state $\mid g⟩$ and the other at an energy $\hslash {\omega }_1$ below the excitation state $\mid e⟩$. Photons from the pump laser will be inelastically scattered at two frequencies ${\omega }_1+\Delta$ and ${\omega }_{eg}$, as shown by wave arrows.

Figure 3

(Color online) Triplet resonances shown in $\mid {\chi }_3^{\left(3\right)}\left(\delta \right)\mid$. In the simulation, the parameters were chosen as ${\gamma }_g∕{\gamma }_2=0.087$, $\mid \Delta \mid ∕{\gamma }_2=13.3$, and $\mid {\Omega }_1\mid ∕{\gamma }_2=1.38$.

Figure 4

(Color online) (a) Experiment: Coincidence counts as a function of the time delay between detected photons [15]. (b) Comparison of $g^{\left(2\right)}\left(\tau \right)$: The red line (with smaller fluctuations) is the experimental data and the blue line (with higher fluctuations) gives the theoretical simulation.