Colloquium: Anomalous statistics of laser-cooled atoms in dissipative optical lattices

Diffusion occurs in numerous physical systems throughout nature, drawing its generality from the universality of the central limit theorem. Approximately a century ago it was realized that an extension to this type of dynamics can be obtained in the form of “anomalous” diffusion, where distributions are allowed to have heavy power-law tails. Owing to a unique feature of its momentum-dependent dissipative friction force, laser-cooled atomic ensembles can be used as a test bed for such dynamics. The interplay between laser cooling and anomalous dynamics bears deep predictive implications for fundamental concepts in both equilibrium and nonequilibrium statistical physics. The high degree of control available in cold-atom experiments allows for the parameters of the friction to be tuned, revealing transitions in the dynamical properties of the system. Rare events in both the momentum and spatial distributions are described by non-normalized states using tools adapted from infinite ergodic theory. This leads to new experimental and theoretical results that illuminate the various features of the system.

Figure 1

Lévy flights are random-walk processes where the jump distances are obtained from heavy-tailed power-law distributions with infinite variance [Eq. (3)]. Depicted is an example of a two-dimensional flight, with the horizontal and vertical axes representing spatial dimensions and a Lévy index $\nu =1.4$. The direction of a given jump is rotated from that of the previous one by a random angle uniformly distributed between 0 and $2\pi$. Inset: depiction of the fractal, self-similar nature of the dynamics, which is similarly characterized by a small number of long flights and a large number of short ones.

Figure 2

Lévy dynamics can arise in the motion of atoms undergoing Sisyphus cooling. Top panel: two counterpropagating, orthogonally linearly polarized laser beams generate a periodic spatial modulation of the polarization. The atomic ground state energy sublevels $M_J=+1/2$ and $-1/2$ are then perturbed by the standing light wave such that the atom travels up and down hills and valleys of potential energy. When the laser frequency is tuned below the atomic resonance frequency, state-changing photon scattering is more probable around the top of the potential hills, where the spontaneously emitted photon has a higher frequency than the absorbed one, illustrated as wiggly and straight yellow lines, respectively. Bottom panel: as a result an atom, moving from left to right in the depicted example along the thicker orange line, loses energy on average due to photon scattering, so cooling becomes efficient. The main control parameter is the modulation depth of the lattice $U_0$. Inset: measurement of the temperature of Sisyphus-cooled ${}^{133}\mathrm{Cs}$ atoms as the lattice depth is varied ([76]), demonstrating a sharp increase of temperature when $U_0/E_R$ decreases. Adapted from [36].

Figure 3

Analysis of the atomic dynamics under the unique Sisyphus friction [Eq. (6)] predicts a non-Gaussian steady state in momentum [Eq. (8)]. Left panel: results of a time-of-flight measurement of the momentum distribution $W(p)$ of an ensemble of ${}^{133}\mathrm{Cs}$ atoms as a function of their momentum, rescaled by the recoil momentum $p_R=\sqrt{2M{E}_R}$. A deep lattice (small red markers, almost coinciding with the thin red line) is compared with a shallow lattice (large black circles). The solid lines represent fits by Eq. (8), giving a Gaussian distribution for the deep lattice (thin red line) and a power law for the shallow lattice (thick black line). Right panel: integrated data from the left panel describing the probability of the momentum being larger than some value $p$. Plotted on a log-log scale, it reveals the power-law nature of the distribution. Adapted from [53].

Figure 4

Top panel: temporal dynamics of the momentum distribution. The rescaled density of momentum vs $p/t^{1/2}$ obtained from Langevin simulations [Eq. (10)] with ${\tilde{U}}_0=2$ for rescaled times $t=t_0\equiv 76.3$, $t=4t_0$, and $t=16t_0$, emphasizing the large momentum of the atom where the scaling is effectively diffusive, as the friction is small. The theoretical curve is the infinite covariant density of Eq. (16), a valid description in the long-time limit. It exhibits a Gaussian-like cutoff that renders the moments finite, unlike those of the formal steady state, which diverge. Adapted from [85]. Bottom panel: the $q$th moment of the momentum as a function of ${\tilde{U}}_0$, showing the complementarity of the two densities in calculating moments of the observables. The steady state [Eq. (11)] is termed Boltzmann-Gibbs since it describes an effective motion in a logarithmic potential (see the text). Adapted from [74].

Figure 5

The nature of the spatial diffusion is revealed by numerical simulation of momentum zero-crossing dynamics. The crossing of zero momentum defines the durations ${\tau }_i$ and jump distances ${\chi }_i$, enabling identification of the connection between the continuous momentum trajectory of the atom and the Lévy walk picture. The jump distances are the areas under the stochastic momentum curve that starts and ends at zero momentum and never crosses in between. The time periods between the zero crossings are indicated by the black arrows at the top. Adapted from [86].

Figure 6

Langevin simulations of the spatial dynamics. Top panel: results of numerical simulations of Eq. (10). Packets of particles starting from a common origin converge in the long-time limit to a self-similar Lévy density that is presented as a solid line [Eq. (22); $\nu =7/6$]. Bottom panel: the simulated distribution also exhibits a second scaling, i.e., data collapse for several measurement times. The convergence is faster for large $x/t^{3/2}$, while for small values deviations from the scaling are found. At large distances the power-law decay of the Lévy density is cut off due to finite time effects. Adapted from [86].

Figure 7

Exponent of the MSD $⟨x^2⟩$ experimentally measured using the dynamics of a weakly harmonically trapped single ${{}^{24}\mathrm{Mg}}^{+}$ ion in a Sisyphus lattice. For deep lattices, the dynamics is diffusive and $⟨x^2⟩\sim t$. As the lattice becomes shallower, a steep rise in the exponent is observed as the dynamics becomes superdiffusive [Eq. (23)]. Inset: time traces of the potential energy of the ion displayed as it moves in the trap. Large rare events are visible in the shallow lattice (top) corresponding to the heavy tails of the distributions, which are compared to the more Gaussian behavior of the deep lattice data (bottom). Adapted from [82].

Figure 8

Self-similarity, Lévy distributions, and superdiffusion. The temporal scaling of the spatial distribution is experimentally measured in one dimension. Top panel: data collapse of the spatial distributions as a function of time, indicated in different colors, for the rescaling transformation $x\to x/t^{1/\nu }$ [Eq. (22) and Fig. 6], with $\nu =1.25$ in a $U_0/E_R=4.8$ lattice. Bottom panel: extracted exponent of the MSD $⟨x^2⟩$. $C$ of Eq. (9) is used as a fitting parameter for Eq. (23), giving a value of $11\pm 1$ for this measurement. The experiment also observed the power-law tails of the spatial distribution predicted by Eq. (22). Adapted from [135].

Figure 9

Anomalous dynamics in momentum and position leads to interesting position-momentum correlations. Top panels: experimental results for the position-momentum correlations as a function of lattice exposure time and lattice depth $U_0/E_R$. At short times the correlations build up and are later quenched at varying rates, depending on the anomalous dynamics. The images on the right are the tomographically measured phase-space probability densities for ballistic expansion (upper panel) and $U_0/E_R=5.5$ (lower panel), at a lattice exposure time of 4.1 ms, showing the effect of the lattice on the development of the correlations. Bottom panel: Langevin simulation (circles) validating the prediction of Eq. (25) depicted as a solid line. Adapted from [4].

Figure 10

Recap of the dynamical phases for various observables. Varying the experimentally accessible control parameter ${\tilde{U}}_0$ defined in Eq. (9) and Fig. 2 reveals transitions in the statistical properties of the system discussed in Sec. 3. Momentum, position, and position-momentum correlations all display sharp transitions between different regimes. Nonphysical divergences are resolved when moments are calculated using the formalism of the infinite covariant density.

Figure 11

Fundamental concepts such as the equipartition theorem are shown to be violated. Steady-state values of the equipartition parameter ${\chi }_H$ [Eq. (40)] experimentally measured for a trapped cloud of ${}^{87}\mathrm{Rb}$ atoms (rescaled by its thermal value, which is $<1$ due to trap anharmonicity), quantifying a nonmonotonic deviation from equipartition. Adapted [3].

Figure 12

Lévy statistics and power-law distributions have meaningful consequences for other atomic systems. One such system is subrecoil laser cooling. Top panel: the fluorescence rate $R(p)$ vanishes for $p\to 0$. Bottom panel: the atoms perform a random walk in momentum space and accumulate in a small interval around $p=0$, where they remain trapped. Adapted from [19].