Blockade-induced resonant enhancement of the optical nonlinearity in a Rydberg medium

We predict a resonant enhancement of the nonlinear optical response of an interacting Rydberg gas under conditions of electromagnetically induced transparency. The enhancement originates from a two-photon process which resonantly couples electronic states of a pair of atoms dressed by a strong control field. We calculate the optical response for the three-level system by explicitly including the dynamics of the intermediate state. We find an analytical expression for the third-order susceptibility for a weak classical probe field. The nonlinear absorption displays the strongest resonant behavior on two-photon resonance where the detuning of the probe field equals the Rabi frequency of the control field. The nonlinear dispersion of the medium exhibits various spatial shapes depending on the interaction strength. Based on the developed model, we propose a realistic experimental scenario to observe the resonance by performing transmission measurements.

Figure 1

(a) Atomic level structure of Rydberg atoms, which interact via an interaction $V\left(R\right)$, that depends on the interatomic distance $R$. (b) Relevant level scheme in the dressed pair-state basis, as explained in the main text. For $\mathrm{\Delta }=-{\mathrm{\Omega }}_c$ the eigenstate $|{\beta }_{-}\rangle$ moves into two-photon resonance with the ground state $|gg\rangle$.

Figure 2

(a) Energy $\mathrm{\Delta }E$ of the dressed levels $|{\beta }_{-}\rangle$ (solid line), $|{\beta }_{+}\rangle$ (dashed line), and $|{\beta }_0\rangle$ (dashed-dotted) against the ratio $|{\mathrm{\Omega }}_c/\mathrm{\Delta }|$ for positive (green) and negative values (black) of $\mathrm{\Delta }$, respectively. $\mathrm{\Delta }E=0$ corresponds to the two-photon resonance with the ground state, which is met for infinite ($R\to 0$, left) and finite (right) interactions for different $|{\mathrm{\Omega }}_c/\mathrm{\Delta }|$. (b) Resonance position $|{\mathrm{\Omega }}_c{/\mathrm{\Delta }|}_{\text{res}}$ against the inter-atomic distance $R$ for positive (green) and negative (black) single-photon detunings.

Figure 3

Real (purple upper line) and imaginary part (blue lower line) of the nonlinear susceptibility ${\chi }^{\left(3\right)}\left(R\right)$ against the interparticle distance $R$ for various ratios ${\mathrm{\Omega }}_c/\mathrm{\Delta }$ (left column: $\mathrm{\Delta }>0$, right column: $\mathrm{\Delta }<0$). The susceptibility is scaled with a factor of $2g^4{\rho }^2/\left(c{\mathrm{\Omega }}_c^2{\overline{\gamma }}_e\right)$. Plotted with ${\mathrm{\Omega }}_c=2.5{\gamma }_e$ and $|48{\text{S}}_{1/2}\rangle$ as the Rydberg state of ${}^{87}\mathrm{Rb}$ atoms for a constant atomic density distribution with $\rho =2\times {10}^{11}{\mathrm{cm}}^{-3}$. The blockade radii $R_b$ are $\{3.6,2.9,2.4\}\mu \text{m}$ for $|{\mathrm{\Omega }}_c/\mathrm{\Delta }|=\{0.3,1.0,5\}$, respectively.

Figure 4

Enhancement of the nonlinear susceptibility. Real (purple, upper) and imaginary (blue, lower) parts of the nonlinear susceptibility ${\chi }_0^{\left(3\right)}$ for strong interactions ($V\left(R\right)\gg {\mathrm{\Omega }}_c$) against the ratio ${\mathrm{\Omega }}_c/\mathrm{\Delta }$. At the resonance position (${\mathrm{\Omega }}_c=\left|\mathrm{\Delta }\right|$) the real part features a sign change, while the imaginary part is resonantly enhanced. Plotted using ${}^{87}\mathrm{Rb}$ atoms with $|48{\text{S}}_{1/2}\rangle$ as the Rydberg state and ${\mathrm{\Omega }}_c=2.5{\gamma }_e$.

Figure 5

Transmission $T=\mathcal{I}\left(L\right)/{\mathcal{I}}_0$ of the probe field after propagating a distance $L$ as a function of the single-photon detuning $\mathrm{\Delta }$ for ${\mathrm{\Omega }}_p/{\mathrm{\Omega }}_c=0.008,0.02,0.04,0.06,0.08$ (black to blue, from top to bottom), respectively. In the nonlinear regime with high ${\mathrm{\Omega }}_p$, two transmission minima for $\mathrm{\Delta }\approx \pm {\mathrm{\Omega }}_c$ appear as a consequence of the two-body, two-photon resonance. Plotted for $\{{\mathrm{\Omega }}_c,\delta ,\rho ,L\}=\{2{\gamma }_e,0,1\times {10}^{11}{\mathrm{cm}}^{-3},400\mu \mathrm{m}\}$ using Eq. (23).