Enhancing light-atom interactions via atomic bunching

There is a broad interest in enhancing the strength of light-atom interactions to the point where injecting a single photon induces a nonlinear material response. Here we show theoretically that sub-Doppler-cooled two-level atoms that are spatially organized by weak optical fields give rise to a nonlinear material response that is greatly enhanced beyond that attainable in a homogeneous gas. Specifically, in the regime where the intensity of the applied optical fields is much less than the off-resonance saturation intensity, we show that the third-order nonlinear susceptibility scales inversely with atomic temperature and, due to this scaling, can be two orders of magnitude larger than that of a homogeneous gas for typical experimental parameters. As a result, we predict that spatially bunched two-level atoms can exhibit single-photon nonlinearities. Our model is valid for all regimes of atomic bunching and simultaneously accounts for the backaction of the atoms on the optical fields. Our results agree with previous theoretical and experimental results for light-atom interactions that have considered only limited regimes of atomic bunching. For lattice beams tuned to the low-frequency side of the atomic transition, we find that the nonlinearity transitions from a self-focusing type to a self-defocusing type at a critical intensity. We also show that higher than third-order nonlinear optical susceptibilities are significant in the regime where the dipole potential energy is on the order of the atomic thermal energy. We therefore find that it is crucial to retain high-order nonlinearities to accurately predict interactions of laser fields with spatially organized ultracold atoms. The model presented here is a foundation for modeling low-light-level nonlinear optical processes for ultracold atoms in optical lattices.

Figure 1

Counterpropagating optical fields (vacuum wave number $k$ and wave number in medium $k^{\prime })$ are applied to a gas of atoms. Three regimes are depicted: I, homogeneous; II, weak bunching; and III, tight bunching.

Figure 2

Bunching parameter as a function of the depth of the dipole potential wells of the optical lattice.

Figure 3

Atomic density distributions for (a) $b=0.01$ $\left(\left|\zeta \right|=0.2\right)$ and (b) $b=0.86$ $\left(\left|\zeta \right|=4\right)$, with $\Delta <0$ (red solid curve) and $\Delta >0$ (blue dotted curve).

Figure 4

Effective susceptibility as a function of $s$ and $b$ for (a) red detunings $\left(\Delta =-3\right)$ and (b) blue detunings $\left(\Delta =3\right)$. The solid curve is the case $T=3/146$, corresponding to a temperature of 3 $\mu$K and $T_D=146$ $\mu$K for rubidium. The long-dashed line is the case where $T\to \infty$ $\left(b=0\right)$. The triangles represent the Taylor-series expansion (20) and the circles represent the asymptotic expansion (21). The vertical dotted lines correspond to $b=0.2$ and $0.8$, which designate the boundaries between regimes I, II, and III for increasing intensity.

Figure 5

Components of ${\chi }_{\text{eff}}$ as functions of $\zeta$: (a) ${\chi }_{\text{lin}}/{\chi }_{\text{lin}}$ (dashed line) and ${\chi }_{\text{bunching}}/{\chi }_{\text{lin}}$ (solid curve) and (b) ${\chi }_{\text{SN}}/{\chi }_{\text{lin}}s$ (dashed line) and ${\chi }_{\text{bunching+SN}}/{\chi }_{\text{lin}}s$ (solid curve).