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arXiv:cond-mat/0606569 · Physical Review Letters

Critical Temperature of Weakly Interacting Dipolar Condensates

Konstantin Glaum, Axel Pelster, Hagen M. Kleinert, Tilman Pfau

Published Feb 23, 2007• 32 citations
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Abstract

We calculate perturbatively the effect of a dipolar interaction upon the Bose-Einstein condensation temperature. This dipolar shift depends on the angle between the symmetry axes of the trap and the aligned atomic dipole moments, and is extremal for parallel or orthogonal orientations, respectively. The difference of both critical temperatures exhibits most clearly the dipole-dipole interaction and can be enhanced by increasing both the number of atoms and the anisotropy of the trap. Applying our results to chromium atoms, which have a large magnetic dipole moment, shows that this dipolar shift of the critical temperature could be measured in the ongoing Stuttgart experiment.

DOI
10.1103/physrevlett.98.080407

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We calculate perturbatively the effect of a dipolar interaction upon the Bose-Einstein condensation temperature. This dipolar shift depends on the angle between the symmetry axes of the trap and the aligned atomic dipole moments, and is extremal for parallel or orthogonal orientations, respectively. The difference of both critical temperatures exhibits most clearly the dipole-dipole interaction and can be enhanced by increasing both the number of atoms and the anisotropy of the trap. Applying our results to chromium atoms, which have a large magnetic dipole moment, shows that this dipolar shift of the critical temperature could be measured in the ongoing Stuttgart experiment.
We consider a dilute gas of dipole moments in an arbitrary harmonic trap and treat both the short-range, isotropic $\ensuremath{\delta}$ interaction and the long-range, anisotropic dipole-dipole interaction perturbatively. In this way, we calculate the leading shift of the critical temperature with respect to that of an ideal gas as a function of the relative orientation of the dipole moments with respect to the harmonic trap axes. In particular, we determine those magic angles, where the dipolar shift of the Bose-Einstein condensation temperature vanishes. Furthermore, we show for the parameters of the ongoing 52Cr^{52}\mathrm{Cr}52Cr BEC experiment of Pfau and co-workers that this dipolar shift can be enhanced by increasing the number of particles, the geometrical mean trap frequency, and the anisotropy of the trap.
We report measurements of the Bose-Einstein condensate critical temperature shift due to dipolar interactions, employing samples of ultracold erbium atoms which feature significant (magnetic) dipole-dipole interactions in addition to tunable contact interactions. Using a highly prolate harmonic trapping potential, we observe a clear dependence of the critical temperature on the orientation of the dipoles relative to the trap axis. Our results are in good agreement with mean-field theory for a range of contact interaction strengths. This work creates an opportunity for further investigations into beyond-mean-field effects and the finite-temperature phase diagram in the more strongly dipolar regime where supersolid and droplet states emerge.
We have measured the effect of dipole-dipole interactions on the frequency of a collective mode of a Bose-Einstein condensate. At relatively large numbers of atoms, the experimental measurements are in good agreement with zero temperature theoretical predictions based on the Thomas-Fermi approach. Experimental results obtained for the dipolar shift of a collective mode show a larger dependency to both the trap geometry and the atom number than the ones obtained when measuring the modification of the condensate aspect ratio due to dipolar forces. These findings are in good agreement with simulations based on a Gaussian ansatz.
The effects of impurity and Bose-Fermi interactions on the transition temperature of a dipolar Bose–Einstein condensation in trapped Bose-Fermi mixture, by using the two-fluid model, are investigated. The shift of the transition temperature consists of four contributions due to contact, Bose-Fermi, dipole-dipole, and impurity interactions. We will show that in the presence of an anisotropic trap, the Bose-Fermi correction to the shift of transition temperature due to the excitation spectra of the thermal part is independent of anisotropy factor. Applying our results to trapped Bose-Fermi mixtures shows that, by knowing the impurity effect, the shift of the transition temperature due to Bose-Fermi interaction could be measured for isotropic trap (dipole-dipole contributions is zero) and Feshbach resonance technique (contact potential contribution is negligible).
We have measured the relative strength ${\ensuremath{\epsilon}}_{dd}$ of the magnetic dipole-dipole interaction compared with the contact interaction in a dipolar chromium Bose-Einstein condensate. We analyze the asymptotic velocities of expansion of the condensate with different orientations of the atomic magnetic moments. By comparing the experimental results with numerical solutions of the hydrodynamic equations for dipolar condensates, we obtain ${\ensuremath{\epsilon}}_{dd}=0.159\ifmmode\pm\else\textpm\fi{}0.034$. We use this result to determine the sss-wave scattering length $a=(5.08\ifmmode\pm\else\textpm\fi{}1.06\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9})\text{ }\mathrm{m}=(96\ifmmode\pm\else\textpm\fi{}20)\text{ }{a}_{0}$ of 52Cr^{52}\mathrm{Cr}52Cr. This is fully consistent with our previous measurements on the basis of Feshbach resonances and therefore confirms the validity of the theoretical approach used to describe the dipolar Bose-Einstein condensate.