Geometric stability spectra of dipolar Bose gases in tunable optical lattices

We examine the stability of quasi-two-dimensional dipolar Bose-Einstein condensates in the presence of weak optical lattices of various geometries. We find that when the condensate possesses a roton-maxon quasiparticle dispersion, the conditions for stability exhibit a strong dependence both on the lattice geometry and the polarization tilt. This results in rich structures in the system's stability diagram akin to spectroscopic signatures. We show how these structures originate from the mode matching of rotons to the perturbing lattice. In the case of a one-dimensional lattice, some of the features emerge only when the polarization axis is tilted into the plane of the condensate. Our results suggest that the stability diagram may be used as a novel means to spectroscopically measure rotons in dipolar condensates.

Figure 1

Stability spectra for q2D dBECs in a weak lattice with varied polarization tilts and scattering lengths, chosen in such a way that $\omega (k_x,0)$ is similar in each case. The shaded regions correspond to stable, condensed ground states. The experimental parameters are for ${}^{164}$Dy trapped at ${\omega }_t=2\pi \times 10$ kHz with a density $n_{3D}\approx {10}^{15}$ cm${}^{-3}$ and dipole moment $d=10{\mu }_B$. Cases I–IV correspond respectively to polarization tilt $\alpha$ (towards $\widehat{y}$) and scattering length $a_s$ values of ${\{\alpha ,a_s\}}_{\mathrm{I}}=\{0^{\circ },-82a_0\}$, ${\{\alpha ,a_s\}}_{\mathrm{II}}=\{7^{\circ },-79.7a_0\}$, ${\{\alpha ,a_s\}}_{\mathrm{III}}=\{{10}^{\circ },-77.35a_0\}$, and ${\{\alpha ,a_s\}}_{\mathrm{IV}}=\{{15}^{\circ },-71.6a_0\}$. The right-hand panels depict the dispersion relations for these cases, with the blue (solid) lines corresponding to $\omega (k_x,0)$ and the red (dashed) lines corresponding to $\omega (0,k_y)$. For each case, the roton wavelength (for modes parallel to $\widehat{x}$) is approximately ${\lambda }_{\mathrm{rot}}\approx 575$ nm. The vertical dashed lines on the stability spectrum denote ${\lambda }_{\mathrm{rot}}/2$, ${\lambda }_{\mathrm{rot}}$, and $2{\lambda }_{\mathrm{rot}}$, which approximately identify the spectral features.

Figure 2

Depiction of roton modes for the case of zero polarization tilt ($\alpha =0$). Degenerate modes satisfying $k_y=k_y^{\prime }$ and $|k_x-k_x^{\prime }|=k_{\mathrm{L}}$ (indicated by pairs of green dots) experience first-order energy shifts.

Figure 3

Schematic for the alignment of lasers (in the $xy$ plane) used to produce the triangular lattice that we consider.

Figure 4

Stability spectrum for a (zero-polarization-tilt) q2D dBEC subjected to a triangular lattice perturbation. The shaded region corresponds to a stable, condensed ground state. The experimental parameters are identical to those of case I in Fig. 1. The assumed laser wavelength is ${\lambda }_{\mathrm{Las}}=595$ nm. The vertical dotted lines denote the predicted locations of stability dips, based on first-order perturbation theory applied to the Gross-Pitaevskii equation. We have employed piecewise cubic interpolation to smooth the stability boundary.

Figure 5

Predicted locations of stability structures as a function of laser wavelength ${\lambda }_{\mathrm{Las}}$. The symmetric blue lines are solutions to $\sin \gamma ={\lambda }_{\mathrm{Las}}/2{\lambda }_{\mathrm{rot}}$, and the red line is the solution to $\cos \frac{\gamma }{2}={\lambda }_{\mathrm{Las}}/2{\lambda }_{\mathrm{rot}}$. Angles corresponding to the laser wavelength used for Fig. 4 are indicated by green dots, and they define the locations of the vertical lines overlain on the triangle-lattice stability spectrum in Fig. 4.