Bose-Einstein condensation (BEC) of an ideal gas is investigated, beyond the thermodynamic limit, for a finite number $N$ of particles trapped in a generic three-dimensional power-law potential. We derive an analytical expression for the condensation
temperature $T_c$ in terms of a power series in $x_0=epsilon_0/k_BT_c$, where $epsilon_0$ denotes the zero-point energy of the trapping potential. This expression, which applies in cartesian, cylindrical and spherical power-law traps, is given analytically at infinite order. It is also given numerically for specific potential shapes as an expansion in powers of $x_0$ up to the second order. We show that, for a harmonic trap, the well known first order shift of the critical temperature $Delta T_c/T_c propto N^{-1/3}$ is inaccurate when $N leqslant 10^{5}$, the next order (proportional to $N^{-1/2}$) being significant. We also show that finite size effects on the condensation temperature cancel out in a cubic trapping potential, e.g. $V(mathbi{r}) propto r^3$. Finally, we show that in a generic power-law potential of higher order, e.g. $V(mathbi{r}) propto r^alpha$ with $alpha > 3$, the shift of the critical temperature becomes positive. This effect provides a large increase of $T_c$ for relatively small atom numbers. For instance, an increase of about +40% is expected with $10^4$ atoms in a $V(mathbi{r}) propto r^{12}$ trapping potential.
We investigate theoretically an original route to achieve Bose-Einstein condensation using dark power-law laser traps. We propose to create such traps with two crossing blue-detuned Laguerre-Gaussian optical beams. Controlling their azimuthal order $
ell$ allows for the exploration of a multitude of power-law trapping situations in one, two and three dimensions, ranging from the usual harmonic trap to an almost square-well potential, in which a quasi-homogeneous Bose gas can be formed. The usual cigar-shaped and disk-shaped Bose-Einstein condensates obtained in a 1D or 2D harmonic trap take the generic form of a finger or of a hockey puck in such Laguerre-Gaussian traps. In addition, for a fixed atom number, higher transition temperatures are obtained in such configurations when compared with a harmonic trap of same volume. This effect, which results in a substantial acceleration of the condensation dynamics, requires a better but still reasonable focusing of the Laguerre-Gaussian beams.
