Symmetry-breaking thermally induced collapse of dipolar Bose-Einstein condensates

We investigate a Bose-Einstein condensate with additional long-range dipolar interaction in a cylindrically symmetric trap within a variational framework. Compared to the ground state of this system, little attention has as yet been payed to its unstable excited states. For thermal excitations, however, the latter is of great interest, because it forms the “activated complex” that mediates the collapse of the condensate. For a certain value of the $s$-wave scatting length our investigations reveal a bifurcation in the transition state, leading to the emergence of two additional and symmetry-breaking excited states. Because these are of lower energy than their symmetric counterpart, we predict the occurrence of a symmetry-breaking thermally induced collapse of dipolar condensates. We show that its occurrence crucially depends on the trap geometry and calculate the thermal decay rates of the system within leading-order transition state theory with the help of a uniform rate formula near the rank-2 saddle which allows the bifurcation to be smoothly passed.

Figure 1

Contour plots of the potential (3) in dependence of the generalized coordinates $q_x$ and $q_y$ for different values of the scattering length: $a/a_d<a_{\mathrm{crit}}$ (a), $a_{\mathrm{crit}}<a/a_d<a_{\mathrm{pb}}$ (b), $a/a_d\gtrsim a_{\mathrm{pb}}$ (c), and $a/a_d\gg a_{\mathrm{pb}}$ (d). The third coordinate $q_z=\mathrm{const}.$ is fixed to the value corresponding to the cylindrically symmetric excited state. See text for description. The color scale is different in every plot. Black corresponds to low values of $V\left(\mathbf{q}\right)$, and white corresponds to high values of $V\left(\mathbf{q}\right)$.

Figure 2

Scaled decay rate $\tilde{\Gamma }=N^2\Gamma$ of the dipolar BEC in dependence of the scattering length $a/a_d$ and for different temperatures $\tilde{\beta }=N^{-1}\beta$. The lines show the results calculated from the conventional TST rate formula (5) and the dots show the corresponding reaction rate obtained from the uniform rate formula (7). The temperatures $T$ as well as the decay rate $\Gamma$ have been calculated for ${}^{52}$Cr BECs with a particle number of $N=50000$.

Figure 3

Stability diagram of the unstable excited state(s) of the GPE in dependence of the scattering length $a/a_d$ and the trap aspect ratio $\lambda$. There exists no excited state, only the symmetric state, or, in addition, two symmetry-breaking states.