We present a largely analytical theory for two-photon correlations
G(2) between Stokes
(s) and anti-Stokes
(a) photon pairs from an extended medium (amplifier) composed of double-$\ensuremath{\Lambda}$ atoms in counterpropagating geometry. We generalize the parametric coupled equations with quantum Langevin noise given in a beautiful experimental paper of Balic et al. [Phys. Rev. Lett. 94, 183601 (2005)] beyond adiabatic approximation and valid for arbitrary strength and detuning of laser fields. We derive an analytical formula for cross correlation ${G}_{as}^{(2)}=⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{s}^{\ifmmode\dagger\else\textdagger\fi{}}(L){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{a}^{\ifmmode\dagger\else\textdagger\fi{}}(0,\ensuremath{\tau}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{a}(0,\ensuremath{\tau}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{E}}_{s}(L)⟩$ and use it to obtain results that are in good quantitative agreement with the experimental data. Results for
Gas(2) obtained using our coupled equations are in good quantitative agreement with the results using the equations of Balic et al., while perfect agreement is obtained for sufficiently large detuning. We also compute the reverse correlation
Gsa(2) which turns out to be negligibly small and remains classical while the cross correlation violates the Cauchy-Schwartz inequality by a factor of more than a hundred.