Effective Field Theory for Rydberg Polaritons

We develop an effective field theory (EFT) to describe the few- and many-body propagation of one-dimensional Rydberg polaritons. We show that the photonic transmission through the Rydberg medium can be found by mapping the propagation problem to a nonequilibrium quench, where the role of time and space are reversed. We include effective range corrections in the EFT and show that they dominate the dynamics near scattering resonances in the presence of deep bound states. Finally, we show how the long-range nature of the Rydberg-Rydberg interactions induces strong effective $N$-body interactions between Rydberg polaritons. These results pave the way towards studying nonperturbative effects in quantum field theories using Rydberg polaritons.

Figure 1

(a) Rydberg polariton transmission experiment: an atomic cloud is probed with a few-photon beam, focused into a 1D channel, and a classical control field. Under dispersive conditions, the total energy $\hslash \nu$ and number $N$ of probe photons are conserved. (b) Interaction potential $V(r)=C_6/r^6$ for two $100S_{1/2}$ Rydberg states in ${}^{87}\mathrm{Rb}$. The range is given by the blockade radius $r_b$. Inset: level diagram of an interacting atom for different $r$. (c) Dimensionless density $n(z)/n(0)=\exp (-z^2/2{\sigma }_{ax}^2)$ and inverse scattering length $r_b/a(z)$ for ${\sigma }_{ax}=36\mu \mathrm{m}$, $r_b=18\mu \mathrm{m}$, $\mathrm{OD}=25$, $\mathrm{\Omega }/2\pi =5\mathrm{MHz}$, and $\mathrm{\Delta }/2\pi =20\mathrm{MHz}$.

Figure 2

(a) Photon correlation function $g^{(2)}(\tau )$ and (b) phase ${\varphi }_2(\tau )$ (in radians) of the transmitted two-photon state calculated using the EFT (solid lines) and numerical simulations (circles). We took $\mathrm{\Omega }/2\pi =5\mathrm{MHz}$, $\mathrm{\Delta }/2\pi =20\mathrm{MHz}$, $r_b=10\mu \mathrm{m}$, $\mathrm{OD}=10$, ${\sigma }_{ax}=36\mu \mathrm{m}$, $L=4{\sigma }_{ax}$, and $L^{\prime }=2.5{\sigma }_{ax}$. To aid comparison we neglect the decay from the $|p⟩$ and $|s⟩$ states.

Figure 3

Examples of (a) a connected and (b) a disconnected scattering diagram for $N=5$. The diagram in (a) contributes to $V_{\mathrm{eff}}^N$, while (b) does not. (c), (d) Graph representations of (a) and (b), respectively, neglecting the ordering of scattering events. The graph in (c) is a tree graph, which implies that (a) is a lowest order diagram for $V_{\mathrm{eff}}^N$. (e) Cut of the nonperturbative solution for $V_{\mathrm{eff}}^N$ in units of ${\chi }_N^{-1}$ up to $N=4$.