Four-wave mixing in three-level systems: Interference and entanglement

The interference of degenerate four-wave mixing (FWM) in three-level systems is examined in both the classical and quantum regimes with a backward configuration. In classical FWM, we find that the phase difference between two indistinguishable FWM transition paths can be varied by different driving laser parameters, and leads to interference in the amplitude and polarization of the generated conjugate field. In the paired-photon generation case, the interference in the nonlinearity disappears because of the time ordering in biphoton generation. However, because of the slow group velocity at the degenerate frequency and polarization, the biphoton-amplitude interference between two Feynman paths can erase the time-ordering information at the detectors. For small group delay, the biphoton correlation, determined by the third-order nonlinearity, shows antibunching and damped Rabi oscillations. For large group delay, where the biphoton bandwidth is determined by phase matching, we show that the biphoton interference leads to a bunching effect. The feasibility of generating polarization entanglement is also discussed.

Figure 1

(Color online) Four-wave mixing in a three-level system. (a) Experimental setup of the right-angle geometry. (b) Two transition paths that produce the ${\omega }_b$ field.

Figure 2

(Color online) Phases of ${\chi }_{\mathrm{I}}^{\left(3\right)}$ and ${\chi }_{\mathrm{II}}^{\left(3\right)}$ vs the normalized coupling detuning ${\Delta }_c∕{\Delta }_{21}$: ${\Delta }_{21}=2\pi \times 3.036\mathrm{GHz}$, ${\gamma }_{31}={\gamma }_{32}=2\pi \times 3\mathrm{MHz}$, ${\gamma }_{21}=2\pi \times 0.03\mathrm{MHz}$, ${\Omega }_c=3{\gamma }_{31}$, and ${\Delta }_p={\Delta }_{21}+{\Delta }_c$.

Figure 3

(Color online) Polarization interference of FWM in an ideal three-state system. (a) Small-angle collinear experimental geometry. (b) Two FWM paths to generate the conjugate field.

Figure 4

(Color online) Polarization of the conjugate field generated via the scheme shown in Fig. 3. (a) ${\Delta }_c=0$, $\varphi =0$; (b) ${\Delta }_c∕{\Delta }_{21}=0.18$, $\varphi =\pi ∕4$; (c) ${\Delta }_c∕{\Delta }_{21}=0.45$, $\varphi =\pi ∕2$; (d) ${\Delta }_c∕{\Delta }_{21}=1.08$, $\varphi =3\pi ∕4$; (e) ${\Delta }_c∕{\Delta }_{21}=14.18$, $\varphi =0.98\pi$; (f) ${\Delta }_c∕{\Delta }_{21}\to \infty$, $\varphi =\pi$.

Figure 5

(Color online) Biphoton generation in a three-level system: (a) Experimental setup; (b) level configuration; (c) Feynman diagram of biphoton interference.

Figure 6

(Color online) Two-photon correlation ${\mid \Psi \left(\tau \right)\mid }^2$ as a function of time delay $\tau$. The blue dashed line represents the results from the perturbation theory and the red solid line stands for those from the Maxwell coupled operator equations. (a) Small-group-delay regime: $\mathrm{OD}=1.5$, ${\Omega }_c=10{\gamma }_{31}$, ${\tau }_g=1.6\mathrm{ns}$, and $t_r=33.5\mathrm{ns}$; (b) large-group-delay regime: $\mathrm{OD}=30$, ${\Omega }_c=2{\gamma }_{31}$, ${\tau }_g=788\mathrm{ns}$, and $t_r=193\mathrm{ns}$.