Four-Wave Mixing and Biphoton Generation in a Two-Level System

We find that in a two-level system there are two kinds of four-wave mixing processes which destructively contribute to the third-order nonlinear susceptibility. In a paired-photon generation scheme by using a single retroreflected pump beam, these two processes occur with definite time orders. The biphoton temporal correlation, which is measured with a back-to-back geometry in a laser-cooled two-level atomic ensemble, shows a damped Rabi oscillation and photon antibunching. Without suppressing the background of Rayleigh scattering, the upper limit of Cauchy criteria is estimated.

Figure 1

Four-wave mixing (FWM) in a two-level system where the field ${\omega }_4=2{\omega }_p-{\omega }_3$ is generated. (a) Experimental configuration. (b) Two FWM processes which may produce the ${\omega }_4$ field.

Figure 2

Schematic of spontaneous paired-photon generation in a two-level system. (a) Simplified experimental configuration. (b) The parametric process which generates the frequencies at ${\omega }_3$ and ${\omega }_4$.

Figure 3

(a) ${}^{87}\mathrm{Rb}$ D2 line hyperfine energy-level diagram and (b) schematic of the experiment for paired-photon generation in a two-level system. In the presence of a linearly polarized, retroreflected pump beam, counterpropagating paired photons are produced into opposing single-mode fibers after linear polarizers ($P$).

Figure 4

Coincidence counts as a function of the time delay $\tau$ between detected photons. Experimental data ($†$, red) was collected over 200 s. Theoretical curves (solid, black) are scaled by a factor of 0.31, 0.46, 0.25, and 0.35 from (a1) to (b2), respectively. (a) Pump power $250\mu \mathrm{W}$: (a1) $\Delta /2\pi =35.6\mathrm{MHz}$, ${\gamma }_g=0.4{\gamma }_e$; (a2) $\Delta /2\pi =83.6\mathrm{MHz}$, ${\gamma }_g=0.3{\gamma }_e$. (b) Pump power $770\mu \mathrm{W}$: (b1) $\Delta /2\pi =35.6\mathrm{MHz}$, ${\gamma }_g=2{\gamma }_e$; (b2) $\Delta /2\pi =83.6\mathrm{MHz}$, ${\gamma }_g=0.3{\gamma }_e$.

Figure 5

(a) Number of total paired counts for 200 s vs optical depth (OD) with a pump power of $770\mu \mathrm{W}$ and a detuning of $\Delta /2\pi =35.6\mathrm{MHz}$. (b) Maximum of the normalized second-order quantum coherence function $g_{34}^{(2)}(\tau {)}_{\max }$ vs pump detuning $\Delta$. The theoretical curve (solid, black), which is dependant of pump power, is calculated with ${\gamma }_g=0.4{\gamma }_e$.