Influence of the number of atoms in a ring-shaped magneto-optical trap: Observation of bifurcation

Numerical study of magneto-optical traps through a hierarchical tree method

“…The study of the collective behavior of atoms under the action of radiation pressure and particle-particle interactions present in cold magneto-optical traps (MOT's) has been the subject of a number of experimental and theoretical investigations [1][2][3][4][5][6][7]. From the experimental side, a variety of shapes of the atomic spatial distribution have been reported [1][2][3][4]7] as the result of the controlled tuning of the number N of trapped atoms, the misalignment of the laser beams, laser intensity, and magnetic field gradient.…”

“…From the experimental side, a variety of shapes of the atomic spatial distribution have been reported [1][2][3][4]7] as the result of the controlled tuning of the number N of trapped atoms, the misalignment of the laser beams, laser intensity, and magnetic field gradient. Indeed, for six Gaussian laser beams well retroreflected and converging to the center of the MOT, an almost spherical cloud of trapped atoms is observed for N ϳ 10 8 -10 9 , with diameter of about 1 mm (static mode).…”

“…In such a case, a theoretical explanation [3,5] has been successfully given in terms only of the vortex forces (radial component of the radiation pressure), since in the low-density regime the influence of the repulsive atom-atom interaction is not relevant. Interestingly, for higher densities ͑N տ 10 7 ͒ the ring structure becomes unstable, as the atoms start to track unstable trajectories due to the increasing repulsive effective interaction and bifurcates into a double-ring orbital mode [4,7]. In this high-density regime the approach only through the vortex force model, strictly valid for low velocities v and low magnetic fields B, does not provide a suitable description of the spatial atomic distribution.…”

“…Instead, if no expansion in v and B is made, no region of parameters is found that supports the existence of a double ring [5]. As a consequence, the theoretical under-standing of the double-ring mode demands a model [6,7] which also takes into account the repulsive radiation force between pairs of trapped atoms due to the reabsorption of scattered photons (secondary photon scattering by the atoms). Such force presents a Coulomb-like spatial decay F ss ϰ 1/d 2 , where d is the distance between two trapped atoms.…”

“…However, as also known from the astrophysical context [8,9], the problem of a large number N of particles interacting through 1 / d 2 long-range decaying forces leads to N fully coupled ordinary differential equations which are extremely difficult to approach even from the numerical viewpoint. Indeed, the numerical tracking of the position and velocity of each particle through direct integration of the equations of motion (direct method) is restricted to systems with, at most, N ϳ 10 4 [7,10], since in this case the computational time grows as ϳ N 2 , being, therefore, very far from our regime of interest in the MOT problem. Alternatively, a mean-field-like approach, typically presenting ϳ N 3/2 , has also been proposed [7] in which all particles that are not nearest neighbors in a numerical grid are considered to give the same contribution to the resulting effective force, whereas only interactions between nearest-neighbor atoms are treated exactly.…”