Finite-temperature trapped dipolar Bose gas

We develop a finite-temperature Hartree theory for the trapped dipolar Bose gas. We use this theory to study thermal effects on the mechanical stability of the system and density oscillating condensate states. We present results for the stability phase diagram as a function of temperature and aspect ratio. In oblate traps above the critical temperature for condensation, we find that the Hartree theory predicts significant stability enhancement over the semiclassical result. Below the critical temperature, we find that thermal effects are well described by accounting for the thermal depletion of the condensate. Our results also show that density oscillating condensate states occur over a range of interaction strengths that broadens with increasing temperature.

Figure 1

(a) Condensate fraction and (b) density oscillation contrast (see text) for a dipolar BEC in a $\lambda =7$ pancake trap. Hartree results (plus symbols), HFBP results (solid lines), and ideal-gas result (dashed line). HFBP data correspond to results shown in Figs. 5 and 6 of Ref. [18]. Other parameters: $\{{\omega }_{\rho },{\omega }_z\}=2\pi \times \{100,700\}{\mathrm{s}}^{-1}$, $N=16.3\times {10}^3$ ${}^{52}$Cr atoms with contact interactions tuned to zero. $T_c^0=\sqrt[3]{N/\zeta \left(3\right)}\hslash \omega /k_B$ is the ideal condensation temperature, where $\omega =\sqrt[3]{{\omega }_{\rho }^2{\omega }_z}$, and $\zeta \left(\alpha \right)$ is the Riemann $\zeta$ function with $\zeta \left(3\right)\approx 1.202$.

Figure 2

Locating instability: (a) The total number of atoms of the self-consistent Hartree solution vs chemical potential for $\lambda =1/8$, $k_{B}T=40\hslash \omega$, and $C_{\mathrm{dd}}=7\times {10}^{-4}\hslash \omega a_{\mathrm{ho}}^3$ (dashed line, case A), $1.5\times {10}^{-4}\hslash \omega a_{\mathrm{ho}}^3$ (solid line, case B), with $a_{\mathrm{ho}}=\sqrt{\hslash /M\omega }$. Each line terminates at the point of instability and occurs at the respective critical number $N_{\mathrm{crit}}$. (b) Same results as in (a), but plotted against ${\epsilon }_0-\mu$. The dotted line represents the target number, in this case $N=2\times {10}^5$. Density profiles: (c), (d) Solid (dashed) line represents the radial (axial) density $n$; higher curves are near the stability boundary. $\lambda =1$, $N=2\times {10}^5$. (c) $T/T_c^0=0.82$ ($N_0/N\approx 0.43$) and $C_{\mathrm{dd}}/4\pi C_0=\{3.65\times {10}^{-4}$ (gray curves), $1.83\times {10}^{-4}$ (black curves)$\}$. (d) $T/T_c^0=1.27$ and $C_{\mathrm{dd}}/4\pi C_0=\{2.91$ (gray curves), $1.22$ (black curves)$\}$. We have introduced the interaction strength unit $C_0=\hslash \omega a_{\mathrm{ho}}^3/\sqrt[6]{N}$, which is convenient for cases where $N$ is fixed and allows our subsequent results to be directly compared to those in Ref. [15].

Figure 3

Stability regions in DDI-temperature space. Shaded regions indicate stability for each geometry, $\lambda =\{8,4,2,1,1/2,1/8\}$ (top to bottom), the geometric mean trap frequency is fixed, and $N=2\times {10}^5$. Actual data points are represented by symbols, while the shading of the stable regions interpolates to guide the eye; the semiclassical model is given by the solid curves. Error bars represent the 1 $\sigma$ spread in the convergence test (see Appendix B3 for more details).

Figure 4

Stability boundary focusing on the below-$T_c$ behavior (line styles as in Fig. 3). For reference, the ideal finite-size-adjusted critical temperature $T_{c,FS}$ [38] for two geometries ($T_{c,FS}^{\lambda =1}$ and $T_{c,FS}^{\lambda =8}$) is indicated by short vertical lines. The effect of DDIs on $T_c$ was calculated perturbatively in Refs. [39, 40]; however, our results are far outside the perturbative regime.

Figure 5

Stability boundary scaling. The stability boundary results (symbols) have been taken from Fig. 4 for $\lambda =\{8,1,1/8\}$ (top to bottom). The dashed-line prediction is based on a noninteracting $N_0$ scaling (see text) and the solid line uses the $N_0$ calculated from the Hartree solutions.

Figure 6

Biconcave characteristics for $\lambda =8$ and $N=2\times {10}^5$ at finite temperature. (a) Stability diagram with biconcave contrast contours $\{0,0.05,0.1,0.15,0.2,0.25\}$ (bottom to top) added. The solid curves are interpolations between the calculated contour points. The white dotted line marks where we terminate the contours due to the condensate fraction becoming negligibly small. Triangles indicate the stability boundary from Fig. 4. Inset: Magnification of the high-temperature region. (b) Radial densities for phase-space points marked by A and B in (a). A: $T/T_c^0=0.0910$, $C_{\mathrm{dd}}/4\pi C_0=0.00268$, and $N_0/N=1.00$ (thermal depletion $<$$1\%$). B: $T/T_c^0=0.910$, $C_{\mathrm{dd}}/4\pi C_0=0.0291$, and $N_0/N=0.0716$. Solid (dashed) lines represent the total (condensate) density. Insets: corresponding surface contours at 67$\%$ of the peak density.

Figure 7

Schematic of our approach to solving dipolar Hartree equations. (a) The low-energy (discrete) modes are explicitly solved by diagonalizing the Hartree equations, whereas higher-energy modes are treated using a semiclassical approximation. (b) Schematic of the range of the spatial grids used to solve the discrete modes and the large grid used to solve the high-energy semiclassical theory.