Role of noise operators on two-photon correlations in an extended coherent Raman medium

An extended medium driven in a double Raman configuration generates Stokes and anti-Stokes fields that are described by coupled parametric oscillator equations with solutions that depend on input boundary operators and quantum noise operators. We identify the conditions where the boundary operators can be the substitute to the noise operators for describing two-photon cross correlation in forward and backward geometries. These conditions include short sample and small decoherence between ground states ${\gamma }_{bc}$, and they are fulfilled by the spontaneous Raman electromagnetic induced transparency scheme (weak pump with large detuning). We verify the correspondence between the results from boundary and noise operators analytically and show that the correlation due to the boundary operators is typically smaller than that due to the noise operators. In general the noise operators are required to obtain the correct correlation, especially when the control laser field is weak and ${\gamma }_{bc}$ is finite. Explanations for the findings are given based on the physics represented by the boundary operators and noise operators. Similar conclusions are obtained for the Stokes and anti-Stokes intensities.

Figure 1

(Color online) (a) Double Raman scheme for forward geometry (copropagating Stokes and anti-Stokes) where spontaneous Raman-EIT corresponds to ${\Delta }_p={\Delta }_s>>{\Omega }_p$ and ${\Delta }_c={\Delta }_a=0$. (b) Summary of the cases where the noise operators are required for correct cross-correlation $⟨{\widehat{E}}_s^{†}\left(t\right){\widehat{E}}_a^{†}(t+\tau ){\widehat{E}}_a(t+\tau ){\widehat{E}}_s\left(t\right)⟩$. (c) The roles and physical origins of the boundary operators and the noise operators.

Figure 2

(Color online) Correlations with relatively small dipole moment $\wp =5\times {10}^{-30}\mathrm{Cm}$ for three categories: (i) $G_{1\text{atom}}^{\left(2\right)}$ for single Raman-EIT atom, (ii) extended medium in forward geometry without decoherence ${\gamma }_{bc}=0$, and (iii) extended medium in forward geometry with decoherence ${\gamma }_{bc}=0.6{\gamma }_{ac}$. The normalized correlation $g_{as}^{\left(2\right)b}=G_{as}^{\left(2\right)b}(L,\tau )∕{\mathcal{I}}_s^b\left(L\right){\mathcal{I}}_a^b\left(L\right)$ is obtained with a commutation relation for the operator at boundary but without the noise operator and $g_{as}^{\left(2\right)n}=G_{as}^{\left(2\right)n}(L,\tau )∕{\mathcal{I}}_s\left(L\right){\mathcal{I}}_a\left(L\right)$ is obtained with only noise operators and no zero boundary operators. Other parameters are $\Delta =-7.5{\gamma }_{ac}$, ${\Omega }_p=0.2{\gamma }_{ac}$, and $N\sigma L=0.528$ ($L=1.5\mathrm{mm}$ and $N=8\times {10}^{16}{\mathrm{m}}^{-3}$). As the control field increases from (a) ${\Omega }_c={\gamma }_{ac}$ to (b) ${\Omega }_c=4.2{\gamma }_{ac}$, the correlation with the boundary operator gives the correct result, in close quantitative agreement with the normalized correlation using only noise operators.

Figure 3

(Color online) Cross correlation $g_{as}^{\left(2\right)}$ and the reverse correlation $g_{sa}^{\left(2\right)}$ computed with boundary (without noise) operators, noise operators only, and both boundary and noise operators for forward geometry. The four cases involve, with decoherence ${\gamma }_{bc}=0.6{\gamma }_{ac}$, no decoherence ${\gamma }_{bc}=0$, short sample $N\sigma L=1$, and long sample $N\sigma L=11$. Other parameters are $\Delta =-7.5{\gamma }_{ac}$, ${\Omega }_p=0.4{\gamma }_{ac}$, and ${\Omega }_c=4.2{\gamma }_{ac}$. We have computed the entire results for backward geometry as well (based on Ref. 1) and find that there is no noticeable difference from the forward case for the cross correlation and reverse correlation, even for a long sample.

Figure 4

(Color online) Same as Fig. 3 except with a smaller control Rabi frequency ${\Omega }_c=0.2{\gamma }_{ac}$. Again, we do not show the entire results for backward geometry since they are indistinguishable from the above figures.

Figure 5

(Color online) The real and imaginary parts of ${\beta }_s$ and ${\beta }_s$ as a function of Fourier frequency $\nu$ for (a) ${\gamma }_{bc}=0.6{\gamma }_{ac}$, $\Delta =-7.5{\gamma }_{ac}$ and (b) with decoherence ${\gamma }_{bc}=0.6{\gamma }_{ac}$ and larger detuning $\Delta =-30{\gamma }_{ac}$, with ${\Omega }_p=0.4{\gamma }_{ac}$.

Figure 6

(Color online) Intensities of the Stokes and anti-Stokes obtained with noise operators ${\mathcal{I}}_s^n,{\mathcal{I}}_a^n$ and with boundary operators ${\mathcal{I}}_s^b,{\mathcal{I}}_a^b$ for forward geometry. (a) Small optical density $N\sigma L=1$ and without decoherence ${\gamma }_{bc}=0$. (b) Small optical density and with decoherence. (c) Large optical density and without decoherence. (d) Large optical density and with decoherence. Other parameters are $\Delta =-7.5{\gamma }_{ac}$ and ${\Omega }_p=0.4{\gamma }_{ac}$. The intensities without noise operators give quite good qualitative results for zero decoherence even if the optical density is large. For small optical density, the intensities increase rapidly with control field but saturate at a small field. In contrast, for large optical density the intensities drop rapidly before saturation. The onset of saturation is larger when ${\gamma }_{bc}$ is finite.