Nonlocal nonlinear response of thermal Rydberg atoms and modulational instability in an absorptive nonlinear medium

Nonlinear and nonlocal effects are discussed in the interaction of laser fields with thermal Rydberg atoms in an electromagnetically induced transparency configuration. We assume that the system's steady state adiabatically follows the time variation in the dipole-dipole interactions due to the atomic motion and use a continuum description for the atomic medium. Based on these approximations, we obtain an analytical form for the nonlocal nonlinear atomic response of the thermal medium and study it for different parameter cases. We further propose a generalized model to describe the modulational instability (MI) in absorptive nonlinear media, in order to understand the propagation dynamics in the thermal Rydberg medium. Interestingly, this model predicts that at short propagation distances, each wave component exhibits the MI effect in absorptive nonlinear media, unlike in the purely dispersive case.

Figure 1

The three-level scheme for a single atom (a) interacting with counter-propagating probe and control fields (b). The two single-photon transitions $|1\rangle \leftrightarrow |2\rangle$ and $|2\rangle \leftrightarrow |3\rangle$ are driven by the two far-detuned probe and control fields with Rabi frequencies ${\mathrm{\Omega }}_c$ and ${\mathrm{\Omega }}_p$, respectively, while the two-photon transition from $|1\rangle$ to the high-lying Rydberg state $|3\rangle$ is of near resonance.

Figure 2

Scaled absorption against the atomic density at different temperatures. The parameters are chosen as ${\lambda }_p=0.795\mu \text{m}, {\lambda }_c=0.48\mu \text{m}, {\mathrm{\Gamma }}_{21}=2\pi \times 5.75\text{MHz}, {\gamma }_{12}={\mathrm{\Gamma }}_{21}/2, {\gamma }_{13}=0.001{\gamma }_{12}, {\gamma }_c=0.01\left|\mathrm{\Delta }k\right|v_p, {\mathrm{\Delta }}_p={\mathrm{\Delta }}_c=0, {\mathrm{\Omega }}_{p0}=2\pi \times 0.5\text{GHz}, {\mathrm{\Omega }}_c=2\pi \times 10\text{GHz}, C_6=140\text{GHz}\mu {\text{m}}^6$, and $S_t=40\mu \text{m},z_L=1\text{cm}$.

Figure 3

(a) Nonlinear coefficient $C_{\mathrm{nl}}$ and (b) the ratio between its real and imaginary parts $\text{Im}\left[C_{\mathrm{nl}}\right]/\text{Re}\left[C_{\mathrm{nl}}\right]$ against temperature $T$. The inset shows the ratio at lower temperatures, where small values indicate that $\text{Im}\left[C_{\mathrm{nl}}\right]$ can be neglected when $T\to 0$. Parameters are $n_0=1.0\times {10}^{20}{\text{m}}^{-3}, {\mathrm{\Delta }}_p=2\pi \times 1.2\text{GHz}, {\mathrm{\Delta }}_c=-{\mathrm{\Delta }}_p$, and $S_t=5r_c$, with $r_c$ being the blockade radius $r_c=\left(\right|C_6{\mathrm{\Delta }}_p{|/{\mathrm{\Omega }}_c^2)}^{1/6}\simeq 0.5\mu \text{m}$. Other parameters are as described in the caption of Fig. 2.

Figure 4

The real (a) and imaginary (b) parts of the nonlocal response function $K_s\left(\xi \right)$ and the real (c) and imaginary (d) parts of $C_{\text{nl}}{K}_s\left(\xi \right)$ at different temperatures. $C_{\text{nl}}{K}_s\left(0\right)$ as a function of $T$ is plotted in (e) and (f). Parameters are the same as described in the caption of Fig. 3.

Figure 5

Probe field propagation through thermal Rydberg atoms at different temperatures $T=0.01\text{K}$ (a), $T=1\text{K}$ (b), and $T=100\text{K}$ (c). (d) One-dimensional sections with ${\xi }_y=0$ at the end of the propagation for the three cases (a)–(c). The localized spikes arise due to the MI effect. Parameters are $S_t=20r_c, w_p=2$, and ${\zeta }_L=1/16$, which corresponds to $z_L\simeq 127.3\mu \text{m}$. Other parameters are as described in the caption of Fig. 3.

Figure 6

(a) Weighted Fourier spectrum $f\left(\mathbf{k}\right)$ of the output probe field at ${\xi }_y=0$. (b) Spectrum as a function of temperature $T$ for three specific wave components $k\in \{-0.36k_t,0.24k_t,6.59k_t\}$. Parameters are as described in the caption of Fig. 5.

Figure 7

(a) Nonlocal nonlinear response $C_{\text{nl}}{K}_F\left(k_x\right)$ as a function of $k_x$ in momentum space. (b) shows the related $\lambda \left(k_x\right)$ defined in Eq. (42). In (c), numerical results $f\left(k\right)$ are compared to the analytical solutions $|a_j|$ for $k_x=6.59k_t$. As incident probe field, a weakly perturbed plane wave is chosen as defined in the main text. Panel (d) is as (c) except for $k_x=30.0k_t$. Parameters are as described in the caption of Fig. 6 but $T=1.0\mathrm{K}$.

Figure 8

(a) $\text{Im}\left[c_0\right]=\text{Im}\left[C_{\text{nl}}{K}_F\left(0\right)\right]$ as a function of temperature $T$. From the numerical calculations, we find that $\text{Im}\left[c_0\right]\approx 0$ at $T=308.5\text{K}$. For this temperature, the corresponding nonlocal nonlinear response $C_{\text{nl}}{K}_F\left(k_x\right)$ in momentum space and ${\lambda }^{\prime }\left(k_x\right)$ defined in Eq. (46) are shown in (b) and (c), respectively. (d) compares $f\left(k\right)$ and $|a_j|$ for $k_x=6.59k_t$. Parameters are ${\mathrm{\Delta }}_p=2\pi \times 0.8\text{GHz}, {\mathrm{\Omega }}_{p0}=2\pi \times 0.25\text{GHz}, {\mathrm{\Omega }}_c=2\pi \times 5\text{GHz}$; other parameters are chosen as in Fig. 3.

Figure 9

Modulational instability at higher atomic density. Panels (a) and (b) show the real and imaginary parts of ${\lambda }^{\prime }\left(k_x\right)$ for two different atomic densities, respectively. At $n_0=6.0\times {10}^{20}{\text{m}}^{-3}$, a range of $k_x$ having real ${\lambda }^{\prime }\left(k_x\right)$ appears, which relates to MI. Parameters are as described in the caption of Fig. 8.

Figure 10

Evolution of Fourier components $|a_0\left(\zeta \right)|, |a_1\left(\zeta \right)|$, and $|a_2\left(\zeta \right)|$ against propagation distance. The black dots denote $f\left(k\right)$, solid blue lines represent the result from a full numerical solution to Eq. (31), and the red solid lines in the middle column show Eq. (45). Note that the middle and right columns are plotted in logarithmic scale. Parameters are $k_x=16.5k_t, {\zeta }_L^{\prime }={\zeta }_L/25$, and other parameters are as described in the caption of Fig. 9.

Figure 11

Weighted Fourier spectrum $f\left(k\right)$ against the propagation distance for two modes $k_x=0.24k_t$ and $k_x=6.59k_t$. The initial probe field is a Gaussian beam as previously employed for Fig. 5. Parameters are as described in the caption of Fig. 5.