Nonlocal Nonlinear Optics in Cold Rydberg Gases

We present an analytical theory for the nonlinear optical response of a strongly interacting Rydberg gas under conditions of electromagnetically induced transparency. Simple formulas for the third-order optical susceptibility are derived and shown to be in excellent agreement with recent experiments. The obtained expressions reveal strong nonlinearities, which in addition are of highly nonlocal character. This property together with the enormous strength of the Rydberg-induced nonlinearities is shown to yield a unique laboratory platform for nonlinear wave phenomena, such as collapse-arrested modulational instabilities in a self-defocusing medium.

Figure 1

(a) Three-level scheme for isolated atoms, where the atomic ground state $|1⟩$, an intermediate state $|2⟩$, and a highly excited Rydberg state $|3⟩$ are mutually driven by a strong control and a weak probe field with Rabi frequencies ${\Omega }_c$ and ${\Omega }_p$, respectively. (b) In a gas of atoms, the strong van der Waals interaction between atoms in state $|3⟩$ inhibits multiple Rydberg excitations within a blockade radius $R_c$, giving rise to a strongly nonlinear optical response of the medium. The resulting nonlinear beam propagation, for example, leads to modulation instabilities, as shown in (c) for a rubidium $70S_{1/2}$ Rydberg gas with a density of $8\times {10}^{13}{\mathrm{cm}}^{-3}$ and ${\Omega }_p/2\pi =0.35\mathrm{MHz}$, ${\Omega }_c/2\pi =80\mathrm{MHz}$, and $\Delta /2\pi =1.2\mathrm{GHz}$.

Figure 2

Nonlinear transmission of a cold rubidium Rydberg-EIT medium with $|3⟩=|60S_{1/2}⟩$ at two different densities and control Rabi frequencies and for ${\overline{\gamma }}_{12}/2\pi =110\mathrm{kHz}$ and ${\gamma }_{13}/2\pi =220\mathrm{kHz}$ [21]. Up to ${\Omega }_{p0}\approx 0.3{\Omega }_c$, there is good agreement between our low-${\Omega }_p$ prediction Eqs. (5, 6) (solid line) and the experimental data [21, 37] (symbols). The dashed lines neglect the drop in absorption due to attenuation and averaging over the initial transverse beam profile.

Figure 3

(a) Effective photon-photon interaction potentials, introduced in Eq. (8). (b) Growth rate ${\Gamma }_{\mathrm{MI}}$ of intensity modulations with wave number $k$ for defocusing nonlinearities of different strengths ${\Omega }^2\alpha$. The dashed lines show the corresponding imaginary part, while the solid lines correspond to the real part of ${\Gamma }_{\mathrm{MI}}$. The critical value of ${\Omega }^2{\alpha }_{\mathrm{MI}}=50.06$ marks the onset of a modulational instability within a narrow window of wave numbers indicated by the gray shaded area for ${\Omega }^2\alpha =80$.

Figure 4

Output beam profile $|\Omega ({\mathbf{r}}_{\perp }){|}^2$ for ${\Omega }_{p0}/2\pi =0.35\mathrm{MHz}$, ${\Omega }_c/2\pi =80\mathrm{MHz}$, and $\Delta /2\pi =1.2\mathrm{GHz}$ after propagation over $l=210\mu \mathrm{m}$ through a $\mathrm{Rb}(70S_{1/2})$ EIT medium at three different densities (a) $4\times {10}^{13}{\mathrm{cm}}^{-3}$, (b) $5.5\times {10}^{13}{\mathrm{cm}}^{-3}$, and (c) $8\times {10}^{13}{\mathrm{cm}}^{-3}$. For the color coding each distribution has been normalized by the actual maximum intensity $|{\Omega }_{\max }{|}^2$.

Figure 5

Intensity profiles $|\Omega ({\mathbf{r}}_{\perp }){|}^2$ for a $\mathrm{Sr}({50}^1{S}_0)$ EIT medium with ${\Omega }_{p0}/2\pi =0.3\mathrm{MHz}$, ${\Omega }_c/2\pi =15\mathrm{MHz}$, and $\Delta /2\pi =3.2\mathrm{GHz}$ at two different densities of (a),(c),(e) $8\times {10}^{11}{\mathrm{cm}}^{-3}$ and (b),(d),(f) $1.2\times {10}^{12}{\mathrm{cm}}^{-3}$. (a),(b) show the stable soliton solutions, while (c)–(f) show the compressed output intensity profile after a propagation length $l=240\mu \mathrm{m}$ for an input beam with (c),(d) $\nu =2$ and (e),(f) $\nu =6$. The color coding is identical to Fig. 4.