We present the Infinite-covariant density (ICD) for the spatial diffusion of atoms undergoing Sisyphus laser-cooling. We show that this is limit law is complementary in this system, as well as in Levy-walks to Levys generalized central limit theorem. 

The semi-classical theory of laser cooled atoms is well developed and was used to find several anomalous effects. Here our motivation is a recent experiment by Davidson et al. from the Weizmann institute (Sagi et al., PRL, 2012), who showed that the spatial probability density function (PDF) $P_t(x)$ of cold Rubidium Rb87 atoms' position $x$ at time $t$, at  a certain regime of parameters, converges at the long time limit to the celebrated Levy distribution. This result was later shown to be related to Levy's generalized central limit theorem (CLT) [Kessler& Baraki, PRL, 2012]. The "problem" with the Levy distribution however, is that naively it may lead to a non-physical expectation, namely a diverging variance $\langle x^2(t)\rangle\rightarrow \infty$. The deviation from the expectations of the CLT and the correct result for the variance were found in a recent paper [Barkai et al., PRX, 2014], where the stochastic dynamics of the cold atoms was modeled as a Levy-Walk process, using the concept of constrained Langevin trajectories, called excursions. 

Using this model for Levy-walks, we find the ICD as a new limit function of the PDF $P_t(x)$, valid at large time $t\rightarrow\infty$ under appropriate scaling. This limit law is complementary to Levy's CLT, in the sense that the ICD provides correct expectation for all the moments of the system, including the variance, but it is non-normalizable. The normalization is insured by the Levy distribution. The two limit function are related to the bi-scaling behavior of the same PDF.

The Sisyphus cooling system provides a unique opportunity to test the predictions of Infinite-ergodic theory in the lab. In addition, our results extend to a wide family of Levy-walks, which in turn are ubiquitous. This suggests that non-normalizable density functions in statistical physics may be describing more physical reality than previously thought. We should start paying attention to them. It might be very interesting!