Scattering resonances and bound states for strongly interacting Rydberg polaritons

We provide a theoretical framework describing slow-light polaritons interacting via atomic Rydberg states. The method allows us to analytically derive the scattering properties of two polaritons. We identify parameter regimes where polariton-polariton interactions are repulsive. Furthermore, in the regime of attractive interactions, we identify multiple two-polariton bound states, calculate their dispersion, and study the resulting scattering resonances. Finally, the two-particle scattering properties allow us to derive the effective low-energy many-body Hamiltonian. This theoretical platform is applicable to ongoing experiments.

Figure 1

(a) Setup for the electromagnetically induced transparency. The probe field couples the atomic ground state $|G\rangle$ to the $p$ level $|P\rangle$ with the single-particle coupling strength $g_0$, while a strong-coupling laser drives the transition between the $p$ level and the Rydberg state $|S\rangle$ with Rabi frequency $\Omega$ and detuning $\delta$. Furthermore, $2\gamma$ denotes the decay rate from the $p$ level. The single-particle coupling $g_0$ is related to the collective coupling $g=\sqrt{n}{g}_0$ with $n$ the particle density. (b) Dispersion relation for the three noninteracting polariton branches for $g=5\delta$, $\Omega =0.2\delta$, and $\gamma =0$.

Figure 2

(a) Low-energy scattering length $a_{1\mathrm{D}}$. For attractive interactions (solid line), we obtain scattering resonances associated with the appearance of additional bound states for increasing interaction strength. The latter is represented by the so-called blockade radius $\xi$. The normalization $\lambda -$ is defined in the text. For repulsive interactions (dashed line), we find a single zero crossing. (b) Two-polariton spectrum for $\Omega \ll \left|\Delta \right|\lesssim g$. For weak interactions $\xi /\lambda -=0.5$ (dashed line), we obtain a single bound state below the continuum of scattering states, whereas for strong interactions $\xi /\lambda -=5$ (solid lines), we observe the existence of several bound states.

Figure 3

(a) Illustration of ladder diagrams up to fourth order. The interaction $V\left(r\right)$ is denoted by a wavy line, while the straight lines with an arrow are Green's functions for the three polariton modes $1/(\hslash \omega -{\epsilon }_{\mu }+i\eta )$ and the dots mark the overlap factors $U_{\mu }^s$ and ${\overline{U}}_s^{\mu }$ of the polariton with the Rydberg state. The $T$ matrix includes all diagrams to arbitrary order with all possible intermediate polaritons. (b) Parameter $\zeta (K,\omega )$ measuring the influence of the second pole for $g\gg \Omega$ and $\Omega /\Delta =0.5$. In the low-energy, low-momentum limit, the second pole can be safely neglected, however its influence strongly increases for $Kc\Delta /2g^2\sim 1$.