Bose–Einstein condensation in spherically symmetric traps

Abstract

We present a pedagogical introduction to Bose-Einstein condensation in traps with spherical symmetry, namely the spherical box and the thick shell, sometimes called bubble trap. In order to obtain the critical temperature for Bose-Einstein condensation, we describe how to calculate the cumulative state number and density of states in these geometries, using numerical and analytical (semi-classical) approaches. The differences in the results of both methods are a manifestation of Weyl's theorem, i.e., they reveal how the geometry of the trap (boundary condition) affects the number of the eigenstates counted. Using the same calculation procedure, we analyzed the impact of going from three-dimensions to two-dimensions, as we move from a thick shell to a two-dimensional shell. The temperature range we obtained, for most commonly used atomic species and reasonable confinement volumes, is compatible with current cold atom experiments, which demonstrates that these trapping potentials may be employed in experiments.

Solution of the 3D logarithmic Schrödinger equation with a central potential

“…Requiring that the wave functions are eigenstates of the parity operator restricts the allowed states in the expansion; this in turn determines the nodal structures in the wave functions. The analysis for a central potential will be useful for studying Bose-Einstein condensates in spherically symmetric atom traps [22][23][24][25] as well as spherically symmetric nuclear models [8].…”

Quantum Bubbles in Microgravity

“…Spherically symmetric hollow condensates have a rich low-energy dynamical behavior [14][15][16][17], and the interplay of curvature, nontrivial contact interaction [18], and finitesize give rise to an interesting phase diagram in the thinshell limit [19][20][21]. Moreover, it is expected that dipolar interactions induce anisotropic density profiles [22,23], while for soft-core interactions a clusterization phenomenon is suggested [24].…”

Shell-shaped condensates with gravitational sag: contact and dipolar interactions

“…BECs in shell-shaped potentials open the possibility to investigate condensation and superfluidity phenomena in new topologies: collective modes [18][19][20], self-interference effects [17], thermodynamics of shells and curved manifolds [21,22], quantized vortices [16,23,24], topological transitions in curved systems [25], the dimensional reduction to a ring-shaped condensate [26], and the transition from filled to hollow condensates [27,28], among others. Theoretical work has focused mainly on shell-shaped BECs with contactinteracting atoms rather than with atoms that possess a non-negligible dipolar moment.…”