Roton confinement in trapped dipolar Bose-Einstein condensates

Roton excitations constitute a key feature of dipolar gases, connecting these gases with superfluid helium. We show that the density dependence of the roton minimum results in a spatial roton confinement, particularly relevant in pancake dipolar condensates with large aspect ratios. We show that roton confinement plays a crucial role in the dynamics after roton instability, and that arresting the instability may create a trapped roton gas revealed by confined density modulations. We discuss the local susceptibility against density perturbations, which we illustrate for the case of vortices. Roton confinement is expected to play a key role in experiments.

Figure 1

(Top) Local spectrum of a BEC of $2\times {10}^5$ Er atoms ${\omega }_z=2\pi \times 1$ kHz and $\lambda =40$. Note a minimum (dark red region in the top panel) in both space and momentum. (Bottom) Change of the roton minimum for two different radial positions.

Figure 2

Roton instability for ${10}^5$ Er atoms, with ${\omega }_z=2\pi \times 450$ Hz ($l_z\simeq 0.3$ $\mu$m), $\lambda =30$, $a_i=8.49a_0>a_c=8.48a_0$, and $a_f=0$ (see text). (a) Column density $n^{2D}\left(\rho \right)=\int n\left(\mathbf{r}\right)dz$ showing concentric rings ($s=0$) formed after $t=19$ ms for small initial fluctuations ($\xi ={10}^{-10}$; see text). (e) Modulational instability after $t=15.5$ ms consisting of several $s$ states for large initial fluctuations ($\xi ={10}^{-6}$, see text). (b) and (f) Momentum distribution of (a) and (e), respectively (we have suppressed the large peak at $q=\sqrt{q_x^2+q_y^2}=0$). (c) and (g) Postcollapse dynamics after $t=23$ ms for (a) and $t=19.5$ ms for (e), respectively. (d) Radial cut of (a) (green crosses) and theoretical column density of the form $n^{2D}\left(\rho \right)\propto {(1-{\rho }^2/R^2)}^{3/2}(1+A{J}_0\left(q_r\rho \right)e^{-{\rho }^2/2l_{*}^2})$ (solid black line), with $R\simeq 51l_z$, $A\simeq -0.4$, $q_r{l}_z\simeq 1.25$, and $l_{*}=2^{1/4}{(R/q_r)}^{1/2}\simeq 7.6l_z$. The value of $q_r$ is calculated using the local spectrum picture. (h) Remnant atoms for the cases (a) (solid red line) and (e) (dashed blue line).

Figure 3

Momentum distribution for different times for the same case depicted in Fig. 2a, as it would be revealed by a TOF experiment. In the top panel we show the $q_x,q_y$ momentum distribution (integrated over $q_z$) and in the middle panel the $q_x,q_z$ momentum distribution (integrated over $q_y$). The bottom panel shows the integrated population at different intervals of the radial momentum $q=\sqrt{q_x^2+q_y^2}$.

Figure 4

Same parameters as in Fig. 2a but arresting the collapse after 16 ms with a quench of $a$ up to $8.55a_0>a_c$. Snapshots of the central region right after the quench (a) and 80 ms later (b). Note a clear density modulation confined at the trap center.

Figure 5

Vortex lattice for a BEC of $N={10}^5$ Er atoms at the threshold of the roton instability. Parameters as in Fig. 1 with a rotational frequency $0.3\omega$.