Correlation of photon pairs from the double Raman amplifier: Generalized analytical quantum Langevin theory

We present a largely analytical theory for two-photon correlations $G^{\left(2\right)}$ between Stokes $\left(s\right)$ and anti-Stokes $\left(a\right)$ photon pairs from an extended medium (amplifier) composed of double-$\Lambda$ 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 $G_{as}^{\left(2\right)}=⟨{\widehat{E}}_s^{†}\left(L\right){\widehat{E}}_a^{†}(0,\tau ){\widehat{E}}_a(0,\tau ){\widehat{E}}_s\left(L\right)⟩$ and use it to obtain results that are in good quantitative agreement with the experimental data. Results for $G_{as}^{\left(2\right)}$ 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 $G_{sa}^{\left(2\right)}$ 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.

Figure 1

(Color online) (a) The double Raman scheme produces correlated pairs of Stokes and anti-Stokes photons. The present four-level scheme (instead of three levels) describes the experimental situation in [9]. The coupling of the pump and Stokes fields to level $\mid a⟩$ is negligible since the fields are far detuned from it. The Raman emission doublet (RED) scheme corresponds to ${\Delta }_{p,s}⪢{\gamma }_{ac}$, ${\Omega }_p$, ${\Delta }_{c,a}=0$, and ${\Omega }_c>{\Omega }_p$ where the Stokes photon is generated by the spontaneous Raman process and the anti-Stokes is by the resonant Raman process. Counterpropagating Stokes and anti-Stokes photons amplified into correlated macroscopic quantum fields ${\widehat{E}}_s$ and ${\widehat{E}}_a$ are generated by lasers propagating (b) along the extended medium (laser depletion may be significant) and (c) perpendicular to the extended medium (negligible laser depletion). Also shown are phase-matched diagrams of the four fields.

Figure 2

(Color online) Comparison of the joint detection results computed from the cross correlation, Eqs. (28, 51). (a) Experimental results of Ref. 9. (b) Hybrid theory based on the coupled equations of Balic et al. [9] and our analytical approach to calculate $G^{\left(2\right)}$. Quantitative agreement is obtained when the coefficients ${\xi }_s$ and ${\xi }_{as}$ in [9] are multiplied by $\sqrt{N{\sigma }_{db}}$ and $\sigma$ is replaced by ${\sigma }_{ac}$ as in Ref. 25 to obtain the correct units. (c) Our complete theory. The curves in (b) and (c) are plotted using the experimental parameters of the decay rates and dipole moments [26] for ${}^{87}\mathrm{Rb}$ with optical depth $N{\sigma }_{ac}L=2\left(g_a{\kappa }_a\right)L∕{\gamma }_{ac}=11$ (where we define the cross sections ${\sigma }_{ac}=\frac{{\mid \wp \mid }^2{\omega }_{ac}}{\hslash c{\epsilon }_{\mathrm{o}}{\gamma }_{ac}},{\sigma }_{db}=\frac{{\mid \wp \mid }^2{\omega }_{db}}{\hslash c{\epsilon }_{\mathrm{o}}{\gamma }_{db}}$), weak pump ${\Omega }_p=0.8{\gamma }_{ac}∕2$, detuning $\Delta =-7.5{\gamma }_{ac}$, decoherence ${\gamma }_{bc}=0.6{\gamma }_{ac}$, ${\Omega }_c∕{\gamma }_{ac}=23.4∕2$, $16.8∕2$, $8.4∕2$, $6∕2$, and $4∕2$. The factor of $1∕2$ is due to a different definition of the Rabi frequency in our case. Quantitative agreement is obtained by multiplying $R_c$ by 16 due to the fact that the electric field defined is twice the physical field. We take the effective transverse cross section as $A=\left(\pi w_{\mathrm{o}}^2\right)\sqrt{3∕4}$ where $w_{\mathrm{o}}=w_1∕\sqrt{2}$ is the waist and $2w_1=280\mu \mathrm{m}$ is the diameter at $1∕e^2$ intensity [9] of the pump laser. The factor of $\sqrt{3∕4}$ leads to a smaller effective area by taking into account the transverse variation of the laser beam.

Figure 3

(Color online) Comparison of the real (solid line) and imaginary (dotted line) pairs of the gain and loss ${\mathcal{G}}_{s,a}\left(\nu \right)$ and coupling ${\mathcal{K}}_{s,a}\left(\nu \right)$ coefficients for (a) $\Delta =-7.5{\gamma }_{ac}$ (as used in experiment) and (b) $\Delta =-75{\gamma }_{ac}$ (larger detuning). Other parameters follow from the experiment [9]: ${\Omega }_p=0.8{\gamma }_{ac}∕2$, ${\Omega }_c=8.4{\gamma }_{ac}∕2$, and $N{\sigma }_{ac}L=11$. Cases (i) coefficients of Balic et al. [9], (ii) our coefficients, Eqs. (7, 8), and (iii) our coefficients but $i\nu ∕c$ is neglected.

Figure 4

(Color online) Normalized self-correlations $g_{ff}^{\left(2\right)n}$, cross correlation $g_{as}^{\left(2\right)n}$, and reverse correlation $g_{sa}^{\left(2\right)n}$ with Cauchy-Schwartz cross correlation $g_{sa}^{CS}$ and reverse correlation $g_{as}^{CS}$ for the experimental parameters $N\sigma L=11$, ${\Omega }_p=0.8{\gamma }_{ac}∕2$, $\Delta =-7.5{\gamma }_{ac}$, ${\gamma }_{bc}=0.6{\gamma }_{ac}$, and ${\Omega }_c∕{\gamma }_{ac}=8.4∕2$.