Electromagnetically-induced-transparency-based paired photon generation

We describe the theory of paired photon generation in a double-$\Lambda$ atomic system. Using the Heisenberg-Langevin formalism we evaluate and analyze the spectral generation rates and intensity correlation function for the output fields. Different regimes of parametric down conversion are analyzed. We discuss the influence of the optical depth and Langevin noise fluctuations on the paired photon generation and predict that at the optical depth of 100 or more the contribution of Langevin noise fluctuations is small and therefore Stokes and anti-Stokes photons are generated mostly in pairs. Comparison between the theory and experiments shows good agreement.

Figure 1

(Color online). Energy-level diagram and schematic for spontaneous backward-wave paired photon generation in an atomic cloud formed by double-$\Lambda$-type atoms. In the presence of the pump and coupling lasers phase-matched, counterpropagating Stokes and anti-Stokes photons are generated into opposite directions.

Figure 2

(Color online) Normalized signal-idler intensity correlation function for the ideal SPDC with the condition of a large group delay $L∕V_g$ for a signal photon. $R$ is the paired photon generation rate. (The figure is taken from Ref. 6.)

Figure 3

(Color online) Transmission and phase mismatch as functions of $\omega$.

Figure 4

(Color online) Coupling constant and Raman gain as functions of detuning $\omega$.

Figure 5

Oscillatory regime: (a) Coincidence count rate in a $1\mathrm{ns}$ bin and (b) Stokes spectral generation rate. $\mathcal{N}\sigma L=0.3$, ${\Omega }_c=6{\gamma }_{13}$, and ${\gamma }_{12}=0$.

Figure 6

(Color online) Group delay regime: (a) Coincidence count rate in a $1\mathrm{ns}$ bin and (b) Stokes spectral generation rate. $\mathcal{N}\sigma L=20$, ${\Omega }_c=6{\gamma }_{13}$, and ${\gamma }_{12}=0$.

Figure 7

(Color online) Group delay regime: (a) Coincidence count rate in a $1\mathrm{ns}$ bin and (b) Stokes spectral generation rate. $\mathcal{N}\sigma L=200$, ${\Omega }_c=6{\gamma }_{13}$, ${\Omega }_p=0.3{\gamma }_{13}$, and ${\gamma }_{12}=0$.

Figure 8

Intensity correlation function with the correlation time smaller than the spontaneous decay time, ${\tau }_r<{\tau }_g<1∕\left(2{\gamma }_{13}\right)$. $\mathcal{N}\sigma L=200$, ${\Omega }_c=65{\gamma }_{13}$, ${\gamma }_{12}=0$, and $1∕\left(2{\gamma }_{13}\right)=25\mathrm{ns}$.

Figure 9

(Color online) Experimentally measured coincidence count rate in (a) the oscillatory regime and (b) between the oscillatory and group delay regimes. $\mathcal{N}\sigma L=11$, ${\Omega }_p=0.8{\gamma }_{13}$, $\Delta {\omega }_{14}=7.5{\gamma }_{13}$, and ${\gamma }_{12}=0.6{\gamma }_{13}$. The bin size is $1\mathrm{ns}$. The experimental data and theory are shown with diamonds and solid lines, respectively. The theoretical curves are scaled vertically by a common factor for each set of the curves. (The figure is taken from Ref. 6.)

Figure 10

(Color online) Stokes (solid curve) and anti-Stokes (dashed curve) spectral generation rates at (a) zero dephasing ${\gamma }_{12}=0$ and (b) nonzero dephasing ${\gamma }_{12}=0.6{\gamma }_{13}$. $\mathcal{N}\sigma L=10$, ${\Omega }_c=6{\gamma }_{13}{\Omega }_p=2.4{\gamma }_{13}$, $\Delta {\omega }_{14}=24{\gamma }_{13}$.

Figure 11

(Color online) Stokes, anti-Stokes (solid curve), and paired (dashed curve) generation rates as a function of the optical depth. ${\Omega }_c=5{\gamma }_{13}$, ${\gamma }_{12}=0$, $\Delta {\omega }_{14}=24{\gamma }_{13}$, and ${\Omega }_p∕\Delta {\omega }_{14}=0.1$. At an optical depth of 100, the paired rate reaches 90% of the Stokes rate.