Spontaneous parametric down-conversion in a three-level system

We have studied the space-time entangled Stokes-anti-Stokes photons generated from a three-level system. In the presence of counterpropagating pump and control lasers, paired Stokes and anti-Stokes fields are spontaneously emitted into opposite directions. The two-photon wave packet is generally a convolution of the phase matching and the third-order nonlinear susceptibility and the feature of the joint temporal correlation is determined by either of them. When the phase matching plays a major role in determining the spectral width of biphotons, the two-photon interference exhibits as conventional type-II spontaneous parametric down-conversion (SPDC). However, if the spectral width of paired photons is determined by the third-order nonlinear susceptibility, two types of four-wave mixings (FWMs) destructively interfere and contribute to the two-photon coincidences. In such a case, the coincidence counting rate appears as a damped Rabi oscillation and exhibits the photon antibunching. The oscillation frequency is equal to the effective Rabi frequency. We have also discussed polarization-entanglement generation in a three-state case. It is found that if the phase matching is crucial in determining the properties of paired Stokes and anti-Stokes photons, the generated state is the same as the degenerate type-II SPDC. However, if the spectral width is mainly controlled by the resonant linewidths in the third-order nonlinearity, one can select one FWM to produce polarization-entangled biphotons, but paired photons from the other FWM process is nonpolarization entangled.

Figure 1

(Color online) (a) Four-wave mixing and biphoton generation in a three-level system. A weak pump beam with frequency ${\omega }_p$ and detuning ${\Delta }_p$ is applied to the atomic transition $|1⟩\to |3⟩$ and a strong control field with frequency ${\omega }_c$ and detuning ${\Delta }_c$ pumps the transition $|2⟩\to |3⟩$. Paired Stokes and anti-Stokes fields are irradiated at the frequencies ${\omega }_s={\varpi }_s+\delta$ and ${\omega }_{as}={\varpi }_{as}-\delta$, respectively. (b) Schematic of a simplified experimental setup. In the presence of the counterpropagating pump and control beams, paired Stokes and anti-Stokes photons trigger photodetectors $D_1$ and $D_2$ in the backward geometry.

Figure 2

(Color online) (a) The linear optical response ${\chi }_{as}\left(\delta \right)$ to the anti-Stokes field: the blue solid line represents the real part of the linear susceptibility and the red point line is the imaginary part. (b) The doublet resonances shown in the third-order nonlinear susceptibility $\left|{\chi }_{as}^{\left(3\right)}\left(\delta \right)\right|$ of the anti-Stokes beam.

Figure 3

(Color online) The normalized two-photon coincidence counting rate in two different regimes, where the paired Stokes-anti-Stokes wave packet is determined by (a) the phase matching and (b) the structure of doublet resonances shown in the third-order nonlinearity. In (b), the red solid line represents the control field resonant with its transition and the blue point line is the opposite case.

Figure 4

(Color online) Polarization-entangled Stokes-anti-Stokes generation in a three-state scheme. Two counterpropagating pump and control beams are circularly polarized with ${\sigma }^{+}-{\sigma }^{-}$ configuration.