Photon-photon correlations from a pair of strongly coupled two-level emitters

We investigate two-color photon correlations in the light emitted by two strongly driven, strongly interacting two-level emitters. The correlations are interpreted introducing the collective dressed states picture, which allows us to describe both bunching and antibunching based on the allowed and prohibited transitions. At odds from weakly interacting emitters, the strong interaction lifts the degeneracy of the energy differences between the different states, leading to a temporal breaking of symmetry for the correlations: photons of different frequencies may not be emitted in any order. Finally, we show that most of the virtual processes, which involve pairs of photons, yield nonclassical correlations when the sum of their energies fits any of the interaction-induced sidebands in the emitted spectrum. In particular, depending on the frequency of the emitted photons, correlations strong enough to violate Bell inequalities can appear.

Figure 1

(a) Collective dressed (left) and bare (right) states for two strongly interacting atoms, in the rotating frame of the laser and in the laboratory frame, respectively. (b) Spectrum of two interacting atoms. The colored arrows above the peaks (of the solid blue line) correspond to the transitions depicted in (a). Simulations carried out for two atoms driven with a field of Rabi frequency $\mathrm{\Omega }=30\mathrm{\Gamma }$, with dipole orientation ${\theta }_{12}={cos}^{-1}(1/\sqrt{3})$, and separated by a distance $k{r}_{12}=0.05$ (solid blue line) and $k{r}_{12}=20$ (dashed red line).

Figure 2

(a) Steady-state photon-photon correlations $g_s^{\left(2\right)}({\omega }_1,{\omega }_2)$ for a pair of strongly interacting, strongly driven atoms. (b) Cascade processes involving the emission of two photons, according to the energy levels of the dressed states picture (allowed cascades with solid lines, forbidden processes with dashed/dotted ones; see text). The associated $g_s^{\left(2\right)}({\omega }_1,{\omega }_2)$ is given by the same symbols in (a). Simulations realized for two atoms with (a) $k{r}_{12}=0.05, {\theta }_{12}={cos}^{-1}(1/\sqrt{3}), \mathrm{\Omega }=30\mathrm{\Gamma }$, and ${\mathrm{\Gamma }}_s=\mathrm{\Gamma }$. (c) Time-resolved $g_s^{\left(2\right)}({\omega }_1,{\omega }_2,\tau )$ for the transitions involving photons of frequency $({\omega }_1,{\omega }_2)=({\mathrm{\Delta }}_{13},-{\mathrm{\Delta }}_{23})$ [$+$ symbol in (a) and (b)] with ${\mathrm{\Gamma }}_s=\mathrm{\Gamma }$. Inset: same curve, for a broader time window. Simulations realized for two atoms with $kr=0.006, \theta ={cos}^{-1}(1/\sqrt{3}), \mathrm{\Omega }=250\mathrm{\Gamma }$, and ${\mathrm{\Gamma }}_s=5\mathrm{\Gamma }$.

Figure 3

Steady-state photon-photon correlations $g_s^{\left(2\right)}({\omega }_1,{\omega }_2)$, for comparison with Fig. 2, for (a) a single atom, and for two interacting atoms with (b) $kr=20$, and (c) $kr=0.4$. Simulations realized for $\mathrm{\Omega }=30\mathrm{\Gamma }, {\mathrm{\Gamma }}_s=\mathrm{\Gamma }$, and $\theta ={cos}^{-1}(1/\sqrt{3})$ (two-atom cases).

Figure 4

(a) Two-photon “leapfrog” processes, with the system transiting through virtual dressed levels. (b) Ratio $R_s$ from the Cauchy-Schwarz inequality, and (c) $B_s$ from the Bell inequality, when tuning the frequency of each sensor. Simulations realized for $kr=0.05, \theta ={cos}^{-1}(1/\sqrt{3}), \mathrm{\Omega }=30\mathrm{\Gamma }$, and ${\mathrm{\Gamma }}_s=\mathrm{\Gamma }$.

Figure 5

Ratio $R_s$ from the Cauchy-Schwarz inequality when tuning the frequency of each sensor, for comparison with Fig. 4, for (a) a single atom, and for two interacting atoms with $\theta ={cos}^{-1}(1/\sqrt{3})$ and (b) $kr=20$ and (c) $kr=0.4$. Simulations are realized for $\mathrm{\Omega }=30\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}_s=\mathrm{\Gamma }$.

Figure 6

Quantifier $R_s$ for the Cauchy-Schwarz inequalities (with a violation above the dashed line) for ${\mathrm{\Gamma }}_s=\mathrm{\Gamma }$ (solid blue) and ${\mathrm{\Gamma }}_s=\mathrm{\Gamma }/10$ (dotted red), for the leapfrog lines (a) ${\omega }_1+{\omega }_2={\mathrm{\Delta }}_{12}$, (b) ${\omega }_1+{\omega }_2=0$, and (c) ${\omega }_1+{\omega }_2=2\mathrm{\Omega }$. (d) Quantifier $B_s$ for the Bell inequalities (with a violation above the dashed line) for ${\mathrm{\Gamma }}_s=\mathrm{\Gamma }$ (solid blue) and ${\mathrm{\Gamma }}_s=\mathrm{\Gamma }/10$ (dotted red) in the leapfrog line of ${\omega }_1+{\omega }_2=0$. The vertical lines with a symbol at the top refer to the processes discussed in Figs. 2, 4 and 4. In (c), we show that collective leapfrog lines for two atoms with $kr=0.4$ do not show a CSI violation as a signature of single-atom physics domination. Simulations realized for $\theta ={cos}^{-1}(1/\sqrt{3})$ and $\mathrm{\Omega }=30\mathrm{\Gamma }$, in (a),(b),(d) $kr=0.05$ and (c) $kr=0.4$.