Two-dimensional Bose gas of tilted dipoles: Roton instability and condensate depletion

We predict the effect of the roton instability for a two-dimensional weakly interacting gas of tilted dipoles in a single homogeneous quantum layer. Being typical for strongly correlated systems, the roton phenomena appear to occur in a weakly interacting gas. It is important that in contrast to a system of normal to the layer dipoles, breaking of the rotational symmetry for a system of tilted dipoles leads to the convergence of the condensate depletion even up to the threshold of the roton instability, with mean-field approach being valid. Predicted effects can be observed in a wide class of dipolar systems. We suggest observing predicted phenomena for systems of ultracold atoms and polar molecules in optical lattices, and estimate optimal experimental parameters.

Figure 1

In (a) BEC gas tilted in the $x-z$ plane dipoles in a 1D optical lattice with the harmonic trap in the $z$ direction. The angle $\theta$ to the layer is controllable by an external (electric or magnetic) field $\vec{\Omega }$. The square of spectrum ${\varepsilon }_{\mathbf{p}}^2$ [see Eq. (1)] as a function of 2D momentum $\mathbf{p}=\{p_x,p_y\}$ at the threshold of the instability: (b) surface plot of ${\varepsilon }_{\mathbf{p}}^2$ at $\theta =0$, (c) at $\theta =0,{\varepsilon }_{\mathbf{p}}^2$ reaches zero at the circumference $\left|\mathbf{p}\right|=p_{\mathrm{r}}$; (d) surface plot of ${\varepsilon }_{\mathbf{p}}^2$ at fixed $\theta \ne 0$; (e) if $\theta \ne 0,{\varepsilon }_{\mathbf{p}}^2$ reaches zero in two points $\{0,\pm p_{\mathrm{r}}\}$, arising at $x=0$ due to polarizing in the $x-z$ plane.

Figure 2

(a) The square of the Bogoliubov spectrum (main figure) and the effective interaction potentials (insets) for different tilt angle $\theta$: a stable homogeneous gas at $\theta =0$ (solid); a phase with roton instability at $\theta =\pi /3$ (dot-dashed); a phase with long-wavelength collapse at $\theta =\pi /2$ (dashed). Stability $\alpha -\theta -\gamma$ diagram (main figures) and typical spectrum branches (insets): in (b) $\alpha -\theta$ projection, where at any $\gamma$ (i.e., the diagram is independent on $\gamma )$, the stable homogeneous and long-wavelength collapse phases are shown, whereas in the solid upper triangular region (with the hypotenuse being ${sin}^2\theta =(2+\alpha )/3)$ the stable phase becomes a roton unstable phase if $\gamma$ increases; in (c)–(e) $\alpha -\gamma$ projection for (c) normal to the layer $\left(\theta =0\right)$, (d) tilted $\left(\theta =\pi /4\right)$, and (e) in-plane $\left(\theta =\pi /2\right)$ dipoles; the boundaries on the diagrams are defined via numerical solution of Eq. (8). Dotted line in (b) corresponds to the magic angle ${\theta }_0=arccos(1/\sqrt{3})$, at which the contribution of the dipolar interaction in the plane reduces to zero [see Eq. (2)].

Figure 3

Divergence of condensate depletion (13) for normal to the layer dipoles (14) and, in contrast, its convergence for tilted dipoles (15). In (a) condensate depletion (13) is shown as a function of the dimensionless density $\gamma$: The depletion at $\theta =\pi /6$ with ${\gamma }_c\approx 3.08$ (dashed line), circle (blue) is the value of depletion at the threshold of the instability; the depletion at $\theta \approx 0.07$ with ${\gamma }_c\approx 5.13$ (solid line), circle (red) is the value of depletion at the threshold of the instability; the depletion at $\theta =0$ with ${\gamma }_c\approx 5.17$ (dot-dashed line). In (b) the condensate depletion is shown as a function of tilt angle $\theta$: first (red) circle on $\theta \approx 0.07$; second (blue) circle on $\theta =\pi /6$.