The Bose-Einstein condensation of a dilute gas of rubidium-87 atoms was achieved by cooling a small number of atoms in a magnetic trap. The effective potential of the trap is to lowest order harmonic and under these conditions we estimate the critical temperature and the heat capacity of an ideal gas of Bose particles. Previously this has been estimated within the semiclassical approximation. However, because of the small number of atoms, the resulting
Tc deviates slightly from our result obtained by exact numerical calculations. For the gap of the heat capacity at the transition point the deviation is
20%.
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.
\nBose-Einstein condensates are trapped in elongated magnetic traps made from\nmicro fabricated current conductors. By bringing the atoms close to the\nsurface of the substrate a periodic anomalous magnetic field is observed. It\nis oriented parallel to the conductor and depends on its current. In a second\nexperiment we study the motion of a condensate in an anharmonic wave guide.\nStrong collective excitations are observed which can be very well described by\ntheory. Simulations predict pronounced chaotic motion within the parameter\nrange of the experiment.\n
The dynamics of weakly interacting three-dimensional Bose-Einstein condensates (BECs), trapped in external axially symmetric plus anharmonic distortion potential are studied. Within a variational approach and time-dependent Gross-Pitaevskii equation, the coupled condensate width equations are derived. By modulating anharmonic distortion of the trapping potential, nonlinear features are studied numerically and illustrated analytically, such as mode coupling of oscillation modes, and resonances. Furthermore, the stability of attractive interaction BEC in both repulsive and attractive anharmonic distortion is examined. We demonstrate that a small repulsive and attractive anharmonic distortion is effective in reducing (extending) the condensate stability region since it decreases (increases) the critical number of atoms in the trapping potential.
Bose-Einstein condensation has been achieved in a magnetic surface microtrap with 4 x 10(5) (87)Rb atoms. The strongly anisotropic trapping potential is generated by a microstructure which consists of microfabricated linear copper conductor of widths ranging from 3 to 30 microm. After loading a high number of atoms from a pulsed thermal source directly into a magneto-optical trap the magnetically stored atoms are transferred into the microtrap by adiabatic transformation of the trapping potential. In the microtrap the atoms are cooled to condensation using forced rf-evaporation. The complete in vacuo trap design is compatible with ultrahigh vacuum below 2 x 10(-11) mbar.
We experimentally produce the rubidium Bose—Einstein condensate in an optically plugged magnetic quadrupole trap. A far blue-detuned focused laser beam with a wavelength of 532 nm is plugged in the center of the magnetic quadrupole trap to increase the number of trapped atoms and to suppress the heating. An rf evaporative cooling in the magneto-optical hybrid trap is applied to decrease the atom temperature into degeneracy. The atom number of the condensate is 1.2(0.4) × 10 5 and the temperature is below 100 nK. We also study characteristic behaviors of the condensate, such as phase space density, condensate fraction and anisotropic expansion.