The critical temperature T c of a weakly interacting Bose gas in an isotropic power-low potential is investigated in the mean-field approximation by taking into account the fact that the particle density distribution function appearing in the mean-field depends on the chemical potential. We derive the general formula of the shift of T c from that of the ideal gas to the lowest order of an interaction. In three-dimensional space, we show that the shift of T c changes its sign from a negative value for n < 3 to a positive one for n > 3, where n is the exponent of the power-low potential.
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
Tc in terms of a power series in ${x}_{0}={\ensuremath{\varepsilon}}_{0}/{k}_{B}{T}_{c}$, where ${\ensuremath{\varepsilon}}_{0}$ 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
x0 up to the second order. We show that, for a harmonic trap, the well-known first-order shift of the critical temperature $\ensuremath{\Delta}{T}_{c}/{T}_{c}\ensuremath{\propto}{N}^{\ensuremath{-}1/3}$ is inaccurate when $N\ensuremath{\leqslant}{10}^{5}$, the next order (proportional to ${N}^{\ensuremath{-}1/2}$) being significant. We also show that finite-size effects on the condensation temperature cancel out in a cubic trapping potential, e.g., $V(r)\ensuremath{\propto}{r}^{3}$. Finally, we show that in a generic power-law potential of higher order, e.g., $V(r)\ensuremath{\propto}{r}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}>3$, the shift of the critical temperature becomes positive. This effect provides a large increase of
Tc for relatively small atom numbers. For instance, an increase of about $+$40% is expected with
104 atoms in a $V(r)\ensuremath{\propto}{r}^{12}$ trapping potential.
By using a mean-field approach, based on the Popov approximation, we calculate the temperature dependence of the condensate fraction of an interacting Bose gas of N atoms confined in an anisotropic harmonic trap. For systems interacting with repulsive forces we find a significant decrease of the condensate fraction and of the critical temperature with respect to the predictions of the noninteracting model with the same value of N. An analytic result for the shift of the critical temperature holding to first order in the scattering length is also derived. \textcopyright{} 1996 The American Physical Society.
The critical temperature
Tc of an interacting Bose gas trapped in a general power-law potential $V(\mathbf{x})={\ensuremath{\sum}}_{i}{U}_{i}{\ensuremath{\mid}{x}_{i}\ensuremath{\mid}}^{{p}_{i}}$ is calculated with the help of variational perturbation theory. It is shown that the interaction-induced shift in
Tc fulfills the relation $({T}_{c}\ensuremath{-}{T}_{c}^{0})∕{T}_{c}^{0}={D}_{1}(\ensuremath{\eta})\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}+{D}^{\ensuremath{'}}(\ensuremath{\eta}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{2\ensuremath{\eta}}+\mathcal{O}({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{2})$ with
Tc0 the critical temperature of the trapped ideal gas, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}$ the
s-wave scattering length divided by the thermal wavelength at
Tc, and $\ensuremath{\eta}=1∕2+{\ensuremath{\sum}}_{i}{p}_{i}^{\ensuremath{-}1}$ the potential-shape parameter. The terms ${D}_{1}(\ensuremath{\eta})\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}$ and ${D}^{\ensuremath{'}}(\ensuremath{\eta}){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{2\ensuremath{\eta}}$ describe the leading-order perturbative and nonperturbative contributions to the critical temperature, respectively. This result quantitatively shows how an increasingly inhomogeneous potential suppresses the influence of critical fluctuations. The appearance of the ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{2\ensuremath{\eta}}$ contribution is qualitatively explained in terms of the Ginzburg criterion.