Number Fluctuations of a Dipolar Condensate: Anisotropy and Slow Approach to the Thermodynamic Regime

We present a theory for the number fluctuations of a quasi-two-dimensional (quasi-2D) dipolar Bose-Einstein condensate measured with finite resolution cells. We show that when the dipoles are tilted to have a component parallel to the plane of the trap, the number fluctuations become anisotropic, i.e., depend on the in-plane orientation of the measurement cell. We develop analytic results for the quantum and thermal fluctuations applicable to the cell sizes accessible in experiments. We show that as cell size is increased the thermodynamic fluctuation result is approached much more slowly than in condensates with short range interactions, so experiments would not require high numerical aperture imaging to observe the predicted effect.

Figure 1

(a) Schematic: a Gaussian cell ${\sigma }_{\perp }$ of size $L_x$ by $L_y$ is used to measure atom number in a quasi-2D dipolar BEC. Anisotropy in number fluctuations is revealed by a second cell ${\sigma }_{\parallel }$ identical to ${\sigma }_{\perp }$, but rotated by 90° (offset in location for clarity). The dipoles are polarized along $\widehat{\mathbf{e}}$, which is tilted in the $xz$ plane at an angle of $\alpha$ to the $z$ axis. (b) Fluctuations as a function of cell area for dipolar BECs with $\alpha =\pi /3$, aspect ratio $\lambda =L_y/L_x=0.15$ (red, $\parallel$) and $\lambda =1/0.15$ (blue, $\perp$) and $\alpha =0$ (green) at $T=0$ (dashed) and $T=2\mu /k_B$ (solid). Thin lines are from Eq. (8) at $T=0$ and Eq. (9) at $T=2\mu /k_B$. The vertical black line is where $\min \{L_{x,y}\}=\xi$. Parameters $A$ and $B$ are detailed in Fig. 2. See text for definitions of $A_{\sigma }$ and $\xi$. Insets: schematic of cells in the $xy$ plane.

Figure 2

Fluctuations as a function of cell aspect ratio with $A_{\sigma }=(60\xi {)}^2$ for parameter set $A$ ($\alpha =\pi /3$, $n{g}_s=0.25\hslash {\omega }_z$, $n{g}_d=0.4\hslash {\omega }_z$, black) and $B$ ($\alpha =0$, $n{g}_s=-0.05\hslash {\omega }_z$, $n{g}_d=0.1\hslash {\omega }_z$, green) at (a) $T=0$ with thin lines from Eq. (8) and (b) $T=2\mu /k_B$ with thin lines from Eq. (9).

Figure 3

(a) Density-density correlation function and (b) static structure factors, at $T=0$ (dashed lines) and $T=2\mu /k_B$ (solid lines) for parameter set $A$ defined in Fig. 2. (inset) Bogoliubov dispersion relation and free particle dispersion ${\epsilon }_{\mathbf{k}}$ (gray). Along the $x,k_x$ axis (red) and along the $y,k_y$ axis (blue). Parameter set $B$ results are isotropic and the same as parameter set $A$ along the $k_y$ axis.

Figure 4

(a) Temperature dependence with $A_{\sigma }=(60\xi {)}^2$ and (b) large $A_{\sigma }$ scaling at $T=2\mu /k_B$, of fluctuations as a function of cell area for dipolar BECs with $\alpha =\pi /3$ ($A$), $\lambda =0.15$ (red) and $\lambda =1/0.15$ (blue) and $\alpha =0$ ($B$, green), comparing to a contact gas, i.e., $g_d=0$, with the same $\mu$ ($C$, magenta dashed-dotted). Thin lines are the analytic approximation (9) for $A$ and $B$ and (10) for $C$. $\times$ shows the analytic approximation (8) at $T=0$. Vertical black lines indicate the $T$ values where ${\lambda }_T=L_x$ and $L_y$, and mark the transition from quantum to thermal regimes. Parameters $A$ and $B$ are detailed in Fig. 2.