Thermally activated local collapse of a flattened dipolar condensate

We consider the metastable dynamics of a flattened dipolar condensate. We develop an analytic model that quantifies the energy barrier to the system undergoing local collapse to form a density spike. We also develop a stochastic Gross-Pitaevskii equation theory for a flattened dipolar condensate, which we use to perform finite-temperature simulations verifying the local collapse scenario. We predict that local collapses play a significant role in the regime where rotons are predicted to exist and will be an important consideration for experiments looking to detect these excitations.

Figure 1

Visualization of the density-spike ansatz [see Eq. (7)] illustrating the parameters used, with $c_w={\left({\pi }^{3/4}{\sigma }_{\rho }\sqrt{{\sigma }_z{l}_z}\right)}^{-1}$.

Figure 2

Spike-formation energy surface $E_s(\beta ,{\sigma }_{\rho },{\sigma }_z)$. Results are shown as a function of $\{{\sigma }_{\rho },\beta \}$ for (a) the stable regime ${\nu }_d<{\nu }_s$, with ${\nu }_d=0.75$, ${\nu }_s=1$, and (b) the metastable regime ${\nu }_d>{\nu }_s$, with ${\nu }_d=1.4$, ${\nu }_s=-0.3$. In (a) we set ${\sigma }_z=\sigma =1.22$ for simplicity. In (b), we choose ${\sigma }_z=1.35$, which minimizes the activation energy $E_A$. (c) Spike energy crossing the saddle of the energy surface along the path shown in (b). The activation energy $E_A$ and the value of $\beta$ at the activation point (${\beta }_A$) are indicated.

Figure 3

Phase diagram and metastable energy barrier. The stable and metastable regimes (which includes the roton regime) and regions of instability are indicated. Contours indicate values of the energy barrier $E_A$ in units of $n_0{l}_z^2\hslash {\omega }_z$.

Figure 4

Field density and a typical spike-formation event. The field density ${\left|\Psi \right|}^2$ is shown (a) at $t=15/{\omega }_z$ (prior to spike formation) and (b) at $t=44/{\omega }_z$ (during spike formation). The red circle indicates the spike location. (c) The peak density in the system during the simulations, revealing the sudden formation of a spike at $t\approx 45/{\omega }_z$. The red crosses indicate the two times corresponding to the fields plotted in (a) and (b). (d) The distribution of densities across the simulation cell prior to collapse. The red arrow indicates $n_A=22.3/l_z^2$. The simulation parameters were $T=0.2\hslash {\omega }_z/k_B$, ${\nu }_s=-0.301$, and ${\nu }_d=1.404$.

Figure 5

The probability that a density spike forms within a time interval of $\delta t=5/{\omega }_z$ after a particular peak density $n_{\mathrm{peak}}$ occurs in the simulation. Calculations are for $T=0.2\hslash {\omega }_z/k_B$.

Figure 6

Temperature dependence of the mean peak formation time ${\overline{t}}_s$, plotted here for two different sets of interaction parameters: ${\nu }_s=-0.301$, ${\nu }_d=1.404$ (as in earlier results; circles), and ${\nu }_s=-0.201$, ${\nu }_d=1.354$ (triangles). The linear fits have slopes of $1.25\pm 0.09$ and $4.1\pm 0.4$.