Statistics of scattered photons from a driven three-level emitter in a one-dimensional open space

We derive the statistics of scattered photons from a $\Lambda$- or ladder-type three-level emitter (3LE) embedded in a one-dimensional open waveguide. The weak probe photons in the waveguide are coupled to one of the two allowed transitions of the 3LE, and the other transition is driven by a control beam. This system shows electromagnetically induced transparency, which is accompanied with the Autler-Townes splitting (ATS) at a strong driving by the control beam, and some of these effects have been observed recently. We show that the nature of second-order coherence of the transmitted probe photons near two-photon resonance changes from bunching to antibunching to constant as the strength of the control beam is ramped up from zero to a higher value where the ATS appears.

Figure 1

Schematic of (a) $\Lambda$- and (b) ladder-type three-level emitters whose one of the two allowed transitions is coupled to probe photons by strength $V$ and the other transition is driven by a control beam with Rabi frequency ${\Omega }_c$.

Figure 2

Appearance of EIT at two-photon resonance, $E_k-E_2=-\Delta$ when a weak control beam (${\Omega }_c<\Gamma$) is switched on, and the ATS appears at a relatively strong control beam $({\Omega }_c>\Gamma )$. The splitting between the Autler-Townes doublet is ${\Omega }_c$. The parameters are $\Delta /\Gamma ={\gamma }_2/\Gamma =1/4,{\gamma }_3/\Gamma =1/40$ for a $\Lambda$-type emitter.

Figure 3

Second-order coherence $g^2(x_2-x_1)$ of the transmitted probe photons from a driven $\Lambda$-type emitter at various control beam driving ${\Omega }_c$ (first row) and two-photon detuning $\delta$ (second row). The probe beam is on two-photon resonance $\delta =[E_k-(E_2-\Delta )]=0$ and ${\gamma }_3/\Gamma =1/40$ in the first row, and the control beam strength ${\Omega }_c/\Gamma =3/10$ and ${\gamma }_3/\Gamma =1/8$ in the second row. The other parameters are $\Delta =0,{\gamma }_2/\Gamma =0.31$.

Figure 4

Second-order coherence $g^2(x_2-x_1)$ of the transmitted probe beam from a driven ladder-type emitter at various control beam driving ${\Omega }_c$. The incident probe beam is a coherent-state wave packet with $k_0=E_2-\Delta \gg {\Delta }_k=\Gamma /40,\overline{n}=1$. The other parameters are $\Delta =0$, $\Gamma /2\pi =11$ MHz, ${\gamma }_2/2\pi =3.4$ MHz, and ${\gamma }_3/2\pi =13.8$ MHz.