## Illustrative trajectories of trapped atoms in a single lobe of the circular optical lattice

**Figure 3.** Illustrative trajectories of trapped atoms in a single lobe of the circular optical lattice. (A) An apparent quasi-periodic behaviour in the three spatial directions is observed. This is the most usual trajectory (~85%) for an initial distribution centred at an intense light lobe. (B) An apparent quasi-periodic behaviour in the transverse plane is observed. Note the high frequencies involved in the axial motion. This kind of trajectory appears for ~9% of the atoms when the initial distribution is centred at the beam axis. (C) The trajectory also seems quasi-periodic in the *XY*-plane. However, though dominant frequencies are observed, they are immersed in a board band. The spectral region (10^{3}Γ) is similar in all directions. This kind of trajectory corresponds to about 5% when the initial distribution is centred in an intense light lobe. (D) The trajectory has an irregular character in the three spatial directions. The spectrum is broad and noisy involving higher frequencies in the axial direction. This kind of trajectory appears irrespective of the initial atom distribution centre, but their percentage is greater ~46% if initially it is centred at the beam axis, ~23% when it is centred between two intense light lobes and just 3% for an initial distribution centred at an intense light lobe. The parameters of the beam and its coupling to the atom are *l* = 2, *k*_{⊥} = 0.66 μm^{−1}, *g*_{peak} = 47.5 Γ, δ = 62.54 GHz.

**Abstract**

We characterize the semiclassical dynamics of dilute thermal atom clouds located in three-dimensional optical lattices generated by stationary optical Bessel beams. The dynamics of the cold atoms is explored in the quasi-Hamiltonian regime that arises using laser beams with far-off resonance detuning. Although the transverse structure of Bessel beams exhibits a complex topological structure, it is found that the longitudinal motion along the main propagation axis of the beam is the detonator of a high sensitivity of the atoms' motion to the initial conditions. This effect would not be properly described by bidimensional models. We show that an experimental implementation can be highly simplified by an analysis of the behaviour of the dynamical system under scale transformations. Experimentally feasible signatures of the chaotic dynamics of the atom clouds are also identified.