Two-photon correlation in a cascade amplifier: Propagation effects via a simple model, nonclassical regimes, and validity of neglecting Langevin noise

We study several aspects of two-photon correlation of an optically driven extended medium (amplifier) with parametric down-conversion (PDC) scheme and three-level cascade scheme. The correlation for the PDC scheme is modeled by coupled parametric equations with constant self-coupling and cross coupling coefficients. This provides simple physical insights to how the correlation profile depends on the sign and magnitude of the coefficients in the presence of propagation group delay between signal and idler photons. The results with constant coefficients are related to the results of the driven three-level cascade scheme obtained from quantum Langevin-Maxwell’s equations. The correlation transforms from bunching to antibunching as the pump field increases. Cauchy-Schwartz inequality is violated for all time delay in the case of off-resonant pump, showing nonclassical correlation. We verify that the correlation obtained without noise operators is qualitatively correct regardless of the optical density, especially for large detuning. We also discuss the interesting physics behind antibunching and oscillations found in the reverse correlation.

Figure 1

(Color online) Correlation $G_{21}^{\left(2\right)}\left(\tau \right)$ (normalized) between the idler and signal field of parametric amplifiers without gain or loss [in Eqs. (5, 6)] and (a) with relative group delay $v_g=c∕{10}^3$ [from Eq. (11)] and (b) without relative group delay [from Eq. (12), and $\delta \left(\tau \right)$ is approximated by $\sin \left({\nu }_{\max }\tau \right)∕\tau ]$. (c) Three-level cascade scheme driven by a pump field for an amplifier of length $L$. The dressed state picture for strong resonant pump depicts quantum interference of the idler field due to ac Stark splitting in levels $a$ and $c$, but no interference from the splitting of level $c$. The $n$ denotes the photon number of the pump field.

Figure 2

(Color online) (a) Correlation $G_{21}^{\left(2\right)}\left(\tau \right)$ (arbitrary unit) versus dimensionless time delay $\tau v_g∕L$ for the simple model with zero gain $G_1=G_2=0$. As the cross coupling coefficients increase, the profile deforms with bulging for $K_1\simeq -K_2$ and denting for $K_1\simeq K_2$. Crudely speaking, both situations correspond to bunching and antibunching profiles, respectively. (b) Correlation using the simple model with finite gain and loss. As the cross coupling coefficients increase, the profile deforms with bulging for $K_1\simeq -K_2$ and denting for $K_1\simeq K_2$. (c) The physics of the above plots can be explained by the illustration. The slow group velocity of signal $c_2=v_g=c∕{10}^3$. In the absence of relative group delay $c_1=c$, the gain or loss has no effect on the correlation which remains as a $\delta$ function. The $\alpha =2g_1{\kappa }_1∕{\gamma }_{ab}$ is the absorption coefficient and $L=1\mathrm{mm}$.

Figure 3

(Color online) Two-photon self-, cross, and Cauchy-Schwartz correlations [$g_{ff}^{\left(2\right)}(L,\tau )\doteq G_{ff}^{\left(2\right)}(L,\tau )∕{\mathcal{I}}_f(L{)}^2$ ($f$ is 1 idler, $2$ signal), $g_{21}^{\left(2\right)}(L,\tau )\doteq G_{21}^{\left(2\right)}(L,\tau )∕{\mathcal{I}}_2\left(L\right){\mathcal{I}}_1\left(L\right)$, and $g_{21}^{CS}=G_{21}^{\left(2\right)}∕\sqrt{G_{22}^{\left(2\right)}{G}_{11}^{\left(2\right)}}$, respectively] for the three-level cascade scheme obtained with full quantum Langevin theory for resonant $\Delta =0$ case. We have used ${\Omega }_p=5{\gamma }_{ac}$, ${\gamma }_p={10}^6{\mathrm{s}}^{-1}$, ${\gamma }_a={\gamma }_b={10}^7{\mathrm{s}}^{-1}$, and $N\sigma L=2\left(g_1{\kappa }_1\right)L∕{\gamma }_{ab}$ with the cross section $\sigma =\mid \wp {\mid }^2{\omega }_{ab}∕\hslash c{\epsilon }_o{\gamma }_{ab}$. The correlation oscillates between nonclassical and classical regimes with time delay. It is interesting to see the correlation profile is preserved for large optical density.

Figure 4

(Color online) Same as Fig. 3 except for detuning $\Delta =-10{\gamma }_{ac}$. The nonclassical correlation is larger than resonant case, but its coherence time shrinks with optical density.

Figure 5

(Color online) Correlations (arbitrary unit) from simple model using the coefficients with the signs and magnitudes corresponding to (a) resonant case and (b) off-resonant case. We have used $\alpha =N\sigma =2g_1{\kappa }_1∕{\gamma }_{ab}$, $v_g=c∕{10}^3$, and $L=1\mathrm{mm}$.

Figure 6

(Color online) Comparison of the cross correlations $g_{21}^{\left(2\right)}$ versus Rabi frequency ${\Omega }_p$ and time delay $\tau$ computed based on Ref. 20 using only boundary operators (without noise) $g_{21}^{\left(2\right)b}$ and only both boundary and noise operators $g_{21}^{\left(2\right)}$ (which gives correct results) for resonant pump field $\Delta =0$. The correlation changes monotonically from bunching to antibunching as the pump field increases. The results without noise operators are in good agreement with the results with noise operators for both a) small and b) large optical densities.

Figure 7

(Color online) Same as Fig. 6 except that pump field is off-resonant $\Delta =-20{\gamma }_{ac}$. The correlation is larger for weaker pump field and always shows bunching. The results without noise operators are in agreement with those with noise operators for both a) small and b) large optical densities. This justifies the neglect of noise operators in the theory concerning the generation of photon pairs in the parametric amplifier [3].

Figure 8

(Color online) Reverse correlations $g_{12}^{\left(2\right)}(L,\tau )\doteq G_{12}^{\left(2\right)}(L,\tau )∕{\mathcal{I}}_2\left(L\right){\mathcal{I}}_1\left(L\right)$ versus Rabi frequency ${\Omega }_p$ and time delay $\tau$ for a) resonant pump and b) off-resonant pump with optical density $N\sigma L=1$.