Structure formation during the collapse of a dipolar atomic Bose-Einstein condensate

We investigate the collapse of a trapped dipolar Bose-Einstein condensate. This is performed by numerical simulations of the Gross-Pitaevskii equation and the novel application of the Thomas-Fermi hydrodynamic equations to collapse. We observe regimes of both global collapse, where the system evolves to a highly elongated or flattened state depending on the sign of the dipolar interaction, and local collapse, which arises due to dynamically unstable phonon modes and leads to a periodic arrangement of density shells, disks, or stripes. In the adiabatic regime, where ground states are followed, collapse can occur globally or locally, while in the nonadiabatic regime, where collapse is initiated suddenly, local collapse commonly occurs. We analyze the dependence on the dipolar interactions and trap geometry, the length and time scales for collapse, and relate our findings to recent experiments.

Figure 1

(Color online) (a) Aspect ratio ${\kappa }_0$ of the ground state solutions as a function of ${\epsilon }_{dd}$ for various trap ratios $\gamma$. Presented are the stable TF solutions of Eq. (11) (solid lines), GPE solutions with $k_{dd}=115$ (crosses), and variational solutions for $k_{dd}=115$ (dashed line). (b) Energy landscapes of Eq. (14) for $\gamma =2$ and ${\epsilon }_{dd}=\left(\mathrm{i}\right)$ $-0.8$, (ii) $-0.52$, (iii) 0.8, (iv) 1.025, and (v) 1.4. Light (dark) regions correspond to high (low) energy, and white contours help to visualize the landscapes.

Figure 2

(Color online) Collapse dynamics in a cigar trap $\gamma =0.1$ for ${\epsilon }_{dd}^0=0.8$ and ${\epsilon }_{dd}^f=1.4$. (a) Energy landscape of Eq. (14) at $t=0$. (b) Energy landscape for $t>0$, with the ensuing TF trajectory indicated (green or gray line). Energy is scaled in units of $\hslash {\omega }_x$. (c) ${\lambda }_x\left(t\right)$ and (d) ${\lambda }_z\left(t\right)$ from the full TF equations of motion (black solid line), the simplified case of Eqs. (23, 24) (solid gray line), and GPE simulations for $k_{dd}=80$ (grey dotted line) and 2000 (black dotted line). (e) Evolution of the energy components during GPE simulations with $k_{dd}=80$ (grey lines) and $k_{dd}=2000$ (black lines). Shown are the kinetic $E_k$ (solid line), zero-point $E_{\mathrm{zp}}$ (dotted line), and dipolar $E_d$ (dashed line) energies, renormalized by the total energy $E_{\mathrm{tot}}$.

Figure 3

(Color online) Snapshots of the collapse of the cigar BEC with $k_{dd}=2000$ in the (a) $x\text{−}y$ plane and (b) $y\text{−}z$ plane at (i) $t=0$, (ii) $t=0.47$, and (iii) $t=0.57{\omega }_x^{-1}$.

Figure 4

(Color online) Collapse dynamics in a pancake trap $\gamma =2$ for ${\epsilon }_{dd}^0=0.8$ and ${\epsilon }_{dd}^f=1.4$. (a) Energy landscape of Eq. (14) at $t=0$. (b) Energy landscape for $t>0$, with the TF trajectory indicated [green (gray) line]. Energy is scaled in units of $\hslash {\omega }_x$. (c) ${\lambda }_x\left(t\right)$ from the TF equations of motion (solid line) and the GPE with $k_{dd}=80$ (dotted line). (d) Same for ${\lambda }_z\left(t\right)$. (e) Kinetic $E_k$, zero-point $E_{\mathrm{zp}}$ and dipolar $E_d$ energies from the GPE, renormalized by the total energy $E_{\mathrm{tot}}$.

Figure 5

(Color online) Snapshots of the BEC density during the collapse of Fig. 4 in (a) $x\text{−}y$ plane and (b) $y\text{−}z$ plane at (i) $t=0$, (ii) $t=1{\omega }_x^{-1}$, and (iii) $t=1.5{\omega }_x^{-1}$. Light (dark) regions correspond to high (low) density, while contours in (ii) help to visualize the density structures.

Figure 6

(Color online) Dynamics in a pancake trap $\gamma =5$ under a sudden change from ${\epsilon }_{dd}=0.8$ to 4. (a) ${\lambda }_x\left(t\right)$ and (b) ${\lambda }_z\left(t\right)$ from the TF equations (solid line) and the GPE with $k_{dd}=80$ (dotted line). (c) Kinetic $E_k$, zero-point $E_{\mathrm{zp}}$, and dipolar $E_d$, energies from the GPE simulations. (d)–(e) Density snapshots during the GPE simulations in the (d) $x\text{−}y$ plane and (e) $y\text{−}z$ plane at (i) $t=0$, (ii) 1, and (iii) $1.5{\omega }_x^{-1}$.

Figure 7

(Color online) Collapse dynamics in a cigar trap $\gamma =0.2$ for ${\epsilon }_{dd}^0=-0.2$ and ${\epsilon }_{dd}^f=-4$. (a) ${\lambda }_x\left(t\right)$ and (b) ${\lambda }_z\left(t\right)$ from the TF equations of motion (solid line) and the GPE with $k_{dd}=80$ (dotted line). (c) Kinetic $E_k$, zero-point $E_{\mathrm{zp}}$, and dipolar $E_d$ energies. (d), (e) GPE density in the (d) $x\text{−}y$ plane and (e) $y\text{−}z$ plane at (i) $t=0$, (ii) 4.4, and (iii) $5{\omega }_x^{-1}$.

Figure 8

(Color online) Collapse dynamics in a pancake trap $\gamma =2$ under a sudden change from ${\epsilon }_{dd}=-0.2$ to $-0.8$. (a) ${\lambda }_x\left(t\right)$ and (b) ${\lambda }_z\left(t\right)$ according to the full TF equations of motion (solid line) and GPE simulations with $k_{dd}=80$ (dotted line). (c) Kinetic $E_k$, zero-point $E_{\mathrm{zp}}$, and dipolar $E_d$ energies from the GPE, renormalized by the total energy $E_{\mathrm{tot}}$. (d), (e) Density snapshots in the (d) $x\text{−}y$ and (e) $y\text{−}z$ planes at (i) $t=0$, (ii) 0.42, and (iii) $0.53{\omega }_x^{-1}$.

Figure 9

Length scale of collapse $l_c$ as a function of ${\epsilon }_{dd}^f$ according to GPE simulations (circles) and the predictions of Eqs. (31, 32) for a homogeneous system $A=B=1$ (dashed lines) and for a trapped system $A\ne B\ne 1$ (solid line). (a) The regime ${\epsilon }_{dd}<-0.5$, assuming a cigar-shaped BEC with $\gamma =0.2$, ${\epsilon }_{dd}^0=-0.2$, and $k_{dd}=-80$. For the cigar prediction (solid line) we assume a TF transverse profile for which $B=\sqrt{3∕2}$. (b) The regime ${\epsilon }_{dd}>1$, assuming a pancake geometry $\gamma =5$ with ${\epsilon }_{dd}^0=0.8$ and $k_{dd}=80$. For the pancake prediction (solid line) we assume a TF axial profile for which $A=\sqrt{5∕4}$. Inset: Homogeneous Bogoluibov spectrum of Eq. (28) for ${\epsilon }_{dd}=1.4$ and for $\theta =0$ (dotted line) and $\theta =\pi ∕2$ (dot-dashed line). Note that the error bars in the GPE results arise from the grid discretization.

Figure 10

Collapse time ${\tau }_c$ following a sudden change in ${\epsilon }_{dd}$ according to the TF equations of motion. Various trap ratios are presented. (a) ${\epsilon }_{dd}^0=-0.2$ and ${\epsilon }_{dd}^f<0$. (b) ${\epsilon }_{dd}^0=0.8$ and ${\epsilon }_{dd}^f>0$. For the case of $\gamma =0.1$ we also plot the analytic expression of Eq. (33) (grey dashed line).

Figure 11

Critical trap ratio ${\gamma }_c$ that stabilizes the BEC against collapse as a function of initial interaction strength ${\epsilon }_{dd}^0$, according to the full TF equations of motion. The final interaction strength ${\epsilon }_{dd}^f={10}^4$ is so large that it is effectively the infinite limit.

Figure 12

(Color online) Collapse dynamics in the Stuttgart system under the linear decrease of $a_s$ over time ${\tau }_r=1\mathrm{ms}$. Density in the (a) $x\text{−}y$ and (b) $y\text{−}z$ plane at (i) $t=0$, (ii) $t=3.1$, and (iii) $t=3.7{\omega }_x^{-1}$.

Figure 13

(Color online) Collapse dynamics in the Stuttgart system under the sudden decrease in $a_s$. Density in the (a) $x\text{−}y$ plane and (b) $y\text{−}z$ plane at (i) $t=0$, (ii) $t=1.08$, and (iii) $t=1.16{\omega }_x^{-1}$.