Dynamics of correlations in a quasi-two-dimensional dipolar Bose gas following a quantum quench

We study the evolution of correlations in a quasi-two-dimensional dipolar gas driven out of equilibrium by a sudden ramp of the interactions. On short time scales, rotonlike excitations coherently oscillate in and out of the condensate, giving rise to pronounced features in the time evolution of the momentum distribution, excited fraction, and the density-density correlation function. The evolution of these correlation functions following a quench can thus be used to probe the spectrum of the dipolar gas. We also find that density fluctuations induced by the presence of rotons following the quench, dramatically slows down the rate of spreading of correlations in the system: Near the roton instability, correlations take infinitely long to build up and show deviations from light-cone-like behavior.

Figure 1

(Top) Evolution of the radial momentum distribution $n_{\mathbf{k}}$ in a quasi-$2\mathrm{D}$ dipolar gas ($l_z$ is the width of the cloud in the direction perpendicular to the $2\mathrm{D}$ plane), following a sudden ramp of the dipolar interaction. Momentum distribution develops a peak at the wave vector (black arrow) corresponding to the roton minimum in the inset. The period of the oscillations is proportional to the roton gap. Black curve is the corresponding equilibrium feature for the same interaction where no prominent roton signature is found [37]. Time ($t$) is given in units of $1/({\nu }_{\mathrm{ho}}=2{\hslash }^2/m{l}_z^2)$.

Figure 2

(Top) Coherent evolution of the dimensionless excited fraction ${\tilde{n}}_{\mathrm{ex}}\left(t\right)=l_z^2{n}_{\mathrm{ex}}/2$ for different values of ${\tilde{g}}_f=g_d/g$. To compare the scale on the vertical axis, the total density is $n{l}_z^2\sim 3$ for typical densities and axial trapping potentials. (From bottom to top) ${\tilde{g}}_f=1$ (thin black), $2.1$ (dotted green), $2.3$ (dashed blue), $2.5$ (thick red). (Bottom left) Energy-momentum dispersion curves for the values of ${\tilde{g}}_f$ in the top panel. (Bottom right) Data show numerically extracted values of the roton gap from the oscillations in the top plot (see solid red curve); solid line is the roton gap extracted from the energy momentum dispersion. For $\tilde{g}<2.3$ the dispersion does not have a sharp rotonlike feature.

Figure 3

(Top) Density-density correlation function for different values of $\mathbf{r}$, for a quench to ${\tilde{g}}_f=1$ (left) and ${\tilde{g}}_f=2.6$ (right), respectively. Arrows indicate the feature whose dispersion we study in the bottom plot for different final values of ${\tilde{g}}_f$. (Bottom left) Temporal location of the dip feature (arrow) in the density-density correlation $\left(t_{\mathrm{dip}}\right)$ for different values of $\mathbf{r}$ and different interactions: black ${\tilde{g}}_f=1$, green (dashed) ${\tilde{g}}_f=2$, blue (dotted) ${\tilde{g}}_f=2.2$, red (dashed-dotted) ${\tilde{g}}_f=2.6$. (Bottom right) Data points show the velocity of spreading of correlations in a quasi-$2\mathrm{D}$ gas for quenches to different ${\tilde{g}}_f$. Inset is a zoom-in for strong dipolar interactions $[\tilde{v}=v/(l_z{\nu }_{\mathrm{ho}}/\sqrt{2})]$; the solid line is the characteristic velocity of rotonlike modes $v=2\Delta /\hslash k_{\mathrm{r}}$ for different ${\tilde{g}}_f$.

Figure 4

(Top) Coherent evolution of the excited fraction ${\tilde{n}}_{\mathrm{ex}}\left(t\right)$ for different values of $\eta$ at ${\tilde{g}}_f=g_d/g=2.5$, obtained by self-consistently solving for the condensate fraction using Eq. (3). The red (solid, thick) curve is the non-self-consistent result (assuming $n_0\approx n$), also shown in Fig. 2 of the main text. The green (dashed), blue (dotted), and black (thin) curves correspond to $\eta =0.05,0.1,0.25$, respectively. (Bottom) Same as top with ${\tilde{g}}_f=2.6$.