Stability and structure of an anisotropically trapped dipolar Bose-Einstein condensate: Angular and linear rotons

We study theoretically Bose-Einstein condensates with polarized dipolar interactions in anisotropic traps. We map the parameter space by varying the trap frequencies and dipolar interaction strengths and find an irregular-shaped region of parameter space in which density-oscillating condensate states occur, with maximum density away from the trap center. These density-oscillating states may be biconcave (red-blood-cell-shaped), or have two or four peaks. For all trap frequencies, the condensate becomes unstable to collapse for sufficiently large dipole interaction strength. The collapse coincides with the softening of an elementary excitation. When the condensate mode is density oscillating, the character of the softening excitation is related to the structure of the condensate. We classify these excitations by linear and angular characteristics. We also find excited solutions to the Gross-Pitaevskii equation, which are always unstable.

Figure 1

(a) Stability plot of a dipolar condensate in an anisotropic trap as a function of the anisotropy parameters ${\lambda }_x$ and ${\lambda }_z$ and dipole strength $D$. In the dark-shaded (magenta) region there exist stable density-oscillating condensates (with the maximum density away from the center of the condensate). The light-shaded (cyan) regions are normal ground states with maximum density at trap center. The slice in the ${\lambda }_z$ direction, corresponding to cylindrical symmetry ${\lambda }_x=1$, is displaced from the axes to improve visibility and reproduces Fig. 1 of Ref. [24]. (b)–(e) Stability plots for constant ${\lambda }_z=6$ (b), 6.5 (c), 7 (d), and 8 (e) with the type of state labeled I–V according to the classification given in Sec. 3b. (f) and (g) Stability plots for constant $D=15$ (f) and 30 (g).

Figure 2

Different types of ground-state density profiles. Density slice of the condensate in the $xy$ plane, focusing on the central region where density oscillations can occur. Solutions give examples of the various condensate types (I–V) and are obtained for the parameters I $({\lambda }_x,{\lambda }_z,D)=(0.65,9,89.4)$, II(a) $(0.85,6,20.3)$, II(b) $(0.9,6.5,26.3)$, III $(0.6,6.5,50.1)$, IV $(0.65,6,37.8)$, and V $(1,6.5,24)$. See text in Sec. 3b for a discussion of the ground-state types.

Figure 3

(a)–(f) Condensate density slices in the $xy$ plane for the following parameters: (a) $({\lambda }_x,{\lambda }_z,D)=(0.8,6.5,35.1)$; (b) $(0.9,6.5,26.3)$; (c) $(1,6.5,24)$; (d) $(0.6,6.5,50.1)$; (e) $(0.65,6,37.8)$; (f) $(0.85,6,20.3)$. (g) Projection of upper surface (large $D$) ground-state types in Fig. 1a onto the ${\lambda }_x{\lambda }_z$ plane. The density-oscillating (dark-shaded) region is divided according to the type of ground state therein. The light-shaded (cyan) regions have no stable density-oscillating ground states for any value of $D$. For reference, contours of constant $\Lambda ={\omega }_z/\sqrt{{\omega }_x{\omega }_y}$ are shown (thick lines).

Figure 4

(a) Stability plot of a dipolar condensate as a function of ${\lambda }_x$, with ${\lambda }_z=5.5$. Two-peaked ground states are indicated by the dark (magenta) region. The dotted line indicates the plotting range of the energy of unstable four-peaked solutions shown in (d). (b) Stable two-peaked solution for $({\lambda }_x,{\lambda }_z,D)=(0.65,5.5,20)$. (c) Unstable four-peaked GPE excitation for $(0.85,5.5,20)$. (d) Energy per particle of type-I ground state (solid line) and unstable, four-peaked (type-IV), GPE excitation (dotted line) as a function of $D$ for the same trap parameters as in (c).

Figure 5

Bogoliubov excitation spectrum as a function of dipole strength $D$, for ${\lambda }_x=0.85$, ${\lambda }_z=6$. The softening mode is labeled by the blue dashed line. The colors represent the parity along the $x$ direction (light, red $=$ even), (dark, black $=$ odd). For these parameters the condensate is a type-I state up until near collapse, where it changes to a type-II state [e.g., see the relevant phase diagram Fig. 1b].

Figure 6

Bogoliubov excitation spectrum as a function of dipole strength $D$ for (a) ${\lambda }_x=0.65$, ${\lambda }_z=9$, (b) ${\lambda }_x=0.85$, ${\lambda }_z=6$, and (c) ${\lambda }_x=0.9$, ${\lambda }_z=6.5$; spectra color-coded to indicate $\langle {\widehat{p}}_x^2\rangle$. Contours of roton density fluctuations ($\delta n_1$) in the $xy$ plane for (R1) $D=89.4$ (with type-I condensate), (R2) $D=20.3$ (type II), and (R3) $D=26.3$ (type II), and trap parameters corresponding to (a), (b), and (c), respectively. Thick lines indicate the nodal lines.

Figure 7

Bogoliubov excitation spectrum as a function of dipole strength $D$ for (a) ${\lambda }_x=0.6$, ${\lambda }_z=6.5$, (b) ${\lambda }_x=0.65$, ${\lambda }_z=6$, and (c) ${\lambda }_x=0.8$, ${\lambda }_z=6.5$. Contours of roton density fluctuations ($\delta n_1$) in the $xy$ plane for (R4) $D=50.1$ (with type-III condensate), (R5) $D=37.8$ (type IV), and (R6) $D=35.1$ (type IV), and trap parameters corresponding to (a), (b), and (c), respectively (also see Fig. 6).