Three-Body Interaction of Rydberg Slow-Light Polaritons

We study a system of three photons in an atomic medium coupled to Rydberg states near the conditions of electromagnetically induced transparency. Based on the analytical analysis of the microscopic set of equations in the far-detuned regime, the effective three-body interaction for these Rydberg polaritons is derived. For slow light polaritons, we find a strong three-body repulsion with the remarkable property that three polaritons can become essentially noninteracting at short distances. This analysis allows us to derive the influence of the three-body repulsion on bound states and correlation functions of photons propagating through a one-dimensional atomic cloud.

Figure 1

(a) Level diagram for the Rydberg EIT setup. (b) The two-body contribution $\sum _{i<j}{V}_{\mathrm{eff}}^{(2)}(x_i-x_j)$ to the total interaction potential $U$ in Jacobi coordinates. Energies are expressed in units of $1/|\chi |$, while lengths are expressed in units of the blockade radius $\xi$. For illustration, we choose a real valued $\mathrm{\Delta }$ with $C_6\mathrm{\Delta }<0$, and set $\alpha =1$. (c) The pure three-body interaction $V_{\mathrm{eff}}^{(3)}$. (d) Total interaction potential $U$ with three-body and two-body contributions, which demonstrates that in this regime three polaritons become noninteracting at short distances.

Figure 2

(a) Adiabatic curves ${\mathrm{\Lambda }}_k(\rho )$ for $\alpha =1$ and $\lambda =0.1$. Note, that only partial waves with a difference in angular momentum by a multiple of 6 are coupled to each other. (b) Lowest effective adiabatic potential ${\mathrm{\Delta }}_0(\rho )$ for different values of $\alpha$ and $\lambda =0.1$. The horizontal thin dashed line denotes the energy of the two-body bound state.

Figure 3

Binding energy $E^{(3)}$ of the three-body bound state in units of the corresponding two-body enegy $E^{(2)}$ as a function of $\lambda$. The dotted (red) line corresponds to the situation of pure two-body interaction, while modifications due to the three-body interaction are shown for weak ($\alpha =0.1$, blue dashed line) and strong ($\alpha =1$, black solid line) three-body repulsion. The insets show the wave function of the bound state for $\lambda =1$ for two cases.

Figure 4

Central peak of the connected part of third-order correlation function ${\tilde{g}}^{(3)}$ for $\lambda =0.1$ and a length of the media $R=20\xi$. (a) In the absence of the three-body interaction, ${\tilde{g}}^{(3)}$ shows the characteristic feature of the three-body bound state determined by a $\delta$-function interaction. (b) The full behavior including the strong three-body interaction with $\alpha =1$, demonstrating the characteristic behavior of the three-body bound state in this regime. Note, that ${\tilde{g}}^{(3)}$ approaches zero for large distances.