Dipolar Bose-Einstein condensates at finite temperature

We study a Bose-Einstein condensate of a dilute gas with dipolar interactions, at finite temperature, using the Hartree-Fock-Bogoliubov theory within the Popov approximation. An additional approximation involving the dipolar exchange interaction is made to facilitate the computation. We calculate the temperature dependence of the condensate fraction of a condensate confined in a cylindrically symmetric harmonic trap. We show that the biconcave-shaped condensates found in [Ronen et al. Phys. Rev. Lett. 98, 30406 (2007)] in certain pancake traps at zero temperature are also stable at finite temperature. Surprisingly, the dip in the central density of these structured condensates is actually enhanced at low finite temperatures. We explain this effect.

Figure 1

(Color online) Condensate fraction as a function of temperature in a pancake trap with aspect ratio 1:4 (i.e., ${\omega }_{\rho }∕{\omega }_z=1∕4$): unpolarized, i.e., dipole moment set to zero (dashed line), polarized (solid line), and ideal gas (dotted line). The total number of ${}^{52}\mathrm{Cr}$ atoms is $100000$.

Figure 2

(Color online) Lowest excitation frequencies a function of temperature for a ${}^{52}\mathrm{Cr}$ condensate in a pancake trap with aspect ratio 1:4: polarized (solid lines) and unpolarized (dashed lines). For each angular momentum number $m=0,1,2,3$, we plot only the lowest mode with this $m$.

Figure 3

(Color online) Condensate fraction as a function of temperature in a cigar trap with aspect ratio 4:1: unpolarized (dashed line), polarized (solid line), and ideal gas (dotted line).

Figure 4

(Color online) Lowest excitation frequencies as a function of temperature for a ${}^{52}\mathrm{Cr}$ condensate in a cigar trap with aspect ratio 4:1: polarized (solid lines) and unpolarized (dashed lines). Only modes which are even in the $z$ direction are shown. For each angular momentum number $m=0,1,2,3$, we plot only the lowest even mode with this $m$.

Figure 5

(Color online) Condensate fraction as a function of temperature in a pancake trap with aspect ratio 1:7, for which biconcave structure is formed at $T=0$: polarized (solid line) and ideal gas (dotted line). The total number of ${}^{52}\mathrm{Cr}$ atoms is $16300$. The scattering length is assumed tuned to 0.

Figure 6

(Color online) Biconcave contrast as a function of temperature in a pancake trap with aspect ratio 1:7. Solid line: contrast parameter for the total density. Dashed line: contrast parameter for the condensed part. Inset: illustration of a typical biconcave density profile showing how the biconcave contrast $c$ is defined.

Figure 7

(Color online) Radial density profiles of the condensate density (solid line) and $10\times$ the thermal component density (dashed line). Note that we scaled the thermal component density by a factor of 10 for visual comparison. The trap aspect ratio is 1:7 as in Fig. 6, and the temperature is $T=0.2T_c$, where $T_c$ is the critical temperature.

Figure 8

(Color online) The mean dipolar field due to the thermal component density whose radial profile is shown in Fig. 7.

Figure 9

(Color online) Lowest excitation frequencies as a function of temperature for a ${}^{52}\mathrm{Cr}$ condensate in a pancake trap with aspect ratio 1:7. Solid line: lowest $m=0$ mode. Dashed line: $m=1$ mode. Dash-dotted line: $m=2$ mode. Dotted line: $m=3$ mode.