Depletion and fluctuations of a trapped dipolar Bose-Einstein condensate in the roton regime

We consider the noncondensate density and density fluctuations of a trapped dipolar Bose-Einstein condensate, focusing on the regime where a rotonlike excitation spectrum develops. Our results show that a characteristic peak in the noncondensate density occurs at trap center due to the rotons. In this regime we also find that the anomalous density becomes positive and peaked, giving rise to enhanced density fluctuations. We calculate the noncondensate density in momentum space and show that a small momentum halo is associated with the roton excitations.

Figure 1

The noncondensate density in the $y=0$ plane for a system with (a1) dipolar interactions $D=220$ ($C=0$) and (b1) contact interactions $C=127$ ($D=0$). Insets: The same data in the main subplot as a false color image with white contours added to show the condensate density in each case [also see Figs. 3a and 3b for the condensate density along the $x$ axis for (a1) and (b1), respectively]. The spectrum of excitations is shown mapped to a dispersion for the (a2) dipolar case and (b2) contact case, only showing modes with angular momentum projection $\left|m\right|\le 2$. The dashed line in (b2) is a visual guide to indicate that the excitation energies increase approximately linearly with radial momentum. (c) ${\tilde{n}}_{\mathrm{in}}$ and (d) ${\tilde{n}}_{\mathrm{out}}$ (see text) for the case shown in (a1), with the energy and $k_{\rho }$ range of modes used to construct these shown by the dashed box in (a2). Other parameters are $\lambda =20$ and $T=10\hslash {\omega }_{\rho }/k_B$.

Figure 2

Development of a roton peak in the noncondensate as the dipole interaction increases for $T=10\hslash {\omega }_{\rho }/k_B$. Noncondensate density is shown in the $y=0$ plane for (a1) $D=80$, (b1) $D=140$, and (c1) $D=200$, with respective mapped dispersions in (a2)–(c2). Insets: The noncondensate density with white lines indicating contours of the condensate density. Note: $C=0$ in all these results.

Figure 3

Comparison between densities of (a) a purely dipolar condensate with $D=220$ and (b) a purely contact interaction condensate with $C=127$. Figures show (blue dashed) the normal density $\tilde{n}$, (red dash dot) the anomalous density $\tilde{m}$, (green solid) $\tilde{n}+\tilde{m}$, and (gray solid) the scaled condensate density, all evaluated along the $x$ axis. (c) The development of the fluctuation density $n_F$ (see text) with dipolar strength for a purely dipolar condensate ($D=120$–220 in steps of $\Delta D=20$). The dashed line shows the real part of $n_F$ for the purely contact case of $C=127$. These results are at $T=10\hslash {\omega }_{\rho }/k_B$. (d) The development of the fluctuation density (see text) with temperature for the purely dipolar condensate with $D=220$ ($T=0\text{–10}\hslash {\omega }_{\rho }/k_B$ in steps of $\Delta T=2\hslash {\omega }_{\rho }/k_B$). Other parameters are $\lambda =20$ and $N_0=25\times {10}^3$.

Figure 4

(a) Stability phase diagram in interaction parameter space for atoms in a $\lambda =20$ trap. The shaded region indicates where a stable solution exists. The dashed line indicates the case of purely dipolar interactions ($C=0$) which were primarily considered earlier in the paper, with the particular case of $C=0,D=220$ indicated by a filled circle. The squares indicate two sets of parameters with nonzero contact interaction that we examine. The square labeled (b) is $C=10,D=220$ with results given in (b1) and (b2). The square labeled (c) is $C=10,D=320$ with results given in (c1) and (c2). (b1), (c1) The (blue dashed) normal density $\tilde{n}$, (red dash dot) anomalous density $\tilde{m}$, (green solid) $\tilde{n}+\tilde{m}$, and (gray solid) scaled condensate density, all evaluated along the $x$ axis. (b2), (c2) The mapped dispersions. Other parameters are $\lambda =20$, $N_0=25\times {10}^3$, and $T=10\hslash {\omega }_{\rho }/k_B$.

Figure 5

(a) Noncondensate density in momentum space for a dipolar condensate with $D=220(C=0)$ (blue lines) and a contact interaction condensate with $C=127(D=0)$ (red dashed line). Inset: Effective 2D interaction obtained using the condensate profile ${\tilde{U}}_{2D}$ and analytically using a Gaussian profile ${\tilde{U}}_{2D}^{\mathrm{A}}$ (see the Appendix for details). (b) Total depletion as a function of the temperature for the contact and dipolar systems considered in (a). Other parameters are as in Fig. 1.