Transition states and thermal collapse of dipolar Bose-Einstein condensates

We investigate thermally excited, dipolar Bose-Einstein condensates (BECs). Quasiparticle excitations of the atomic cloud cause density fluctuations which can induce the collapse of the condensate if the interparticle interaction is attractive. Within a variational approach, we identify the collectively excited stationary states of the gas which form transition states on the way to the BEC's collapse. We analyze transition states with different $m$-fold rotational symmetry and identify the one which mediates the collapse. The latter's symmetry depends on the trap aspect ratio of the external trapping potential, which determines the shape of the BEC. Moreover, we present the collapse dynamics of the BEC and calculate the corresponding decay rate using transition-state theory. We observe that the thermally induced collapse mechanism is important near the critical scattering length, where the lifetime of the condensate can be significantly reduced. Our results are valid for an arbitrary strength of the dipole-dipole interaction. Specific applications are discussed for the elements ${}^{52}$Cr, ${}^{164}$Dy, and ${}^{168}$Er with which dipolar BECs have been experimentally realized.

Figure 1

Mean-field energy $E$ of the stationary states of a dipolar BEC for a mean trap frequency $\overline{\gamma }=8000$ and a trap aspect ratio of (a) $\lambda =4$ and (b) $\lambda =7$. Shown are the values obtained from the simple variational ansatz using a single Gaussian wave function ($N_g=1$) and the extended approach using the trial wave functions in Eqs. (8) and (9) ($N_g=6$). The solid and dash-dotted lines represent the ground state of the condensate and the dashed lines represent the collectively excited states. Using the simple variational ansatz, the energy difference between the ground and the excited state increases very rapidly while it is smaller than the linewidth of the plot within the extended approach. The vertical lines indicate the positions of the bifurcations (cf. Fig. 2).

Figure 2

Energy barrier $E^{‡}=E_{\mathrm{ex}}-E_{\mathrm{gs}}$ of a dipolar BEC for a mean trap frequency $\overline{\gamma }=8000$ and a trap aspect ratio of (a) $\lambda =4$ and (b) $\lambda =7$. Both subfigures show the region of the $s$-wave scattering length near the critical scattering length $a_{\mathrm{crit}}$. For $\lambda =4$ and a scattering length above the critical value ($a_{\mathrm{crit}}\approx 0.0619$), there exist only the ground state of the BEC and a collectively excited state with $m=0$ symmetry. For $\lambda =7$, several collectively excited states with different symmetry $m=0,2,3,4$ are present. The one which is involved in the bifurcation at the critical scattering length ($a_{\mathrm{crit}}\approx -0.0016$) is the $m=2$ excited state and it is this state which has the smallest energy barrier in the parameter region where the BEC is stable. (See text for further explanations.)

Figure 3

Collapse dynamics of an excited dipolar BEC at (a) a trap aspect ratio $\lambda =4$ and a scattering length of $a/a_d=6.233\times {10}^{-2}$ as well as (b) $\lambda =7$ and $a/a_d=3.940\times {10}^{-4}$. Shown is the time evolution of the BEC's rms extensions $\sqrt{\langle x^2\rangle }$ and $\sqrt{\langle y^2\rangle }$, respectively, and the excitation energy that is slightly above the energy barrier, which is $E^{‡}=1.0$ for both values of the scattering length given above. The dynamics shows (I) collective oscillations of the atomic cloud, (II) the formation of the quasistationary activated complex (AC), and (III) the collapse of the BEC.

Figure 4

Activation energy $E^{‡}$ for the thermally induced coherent collapse for (a) non-blood-cell-shaped BECs at trap aspect ratios $3\le \lambda \le 4.5$ and (b) blood-cell-shaped BECs at $7\le \lambda \le 8$. In (a), the energy barrier with increasing scattering length becomes smaller the higher the trap aspect ratio is. For blood-cell-shaped BECs in (b), the behavior of the energy barrier with increasing scattering length changes only marginally when varying the trap aspect ratio. We note that for a ${}^{52}$Cr condensate consisting of $N=10000$ bosons, the range of the energy barrier shown ($0\le E^{‡}\le 7$) corresponds to a thermal energy $T=E^{‡}/k_B$ of $0\le T\le 141$nK, which is about 20% of the critical temperature ($T_{\mathrm{c}}=700$ nK [1]). The range of the scattering length is $0.275$ Bohr radii above the critical scattering length in (a) and $0.55$ Bohr radii in (b).

Figure 5

Decay rate due to the thermally induced collapse for (a) non-blood-cell-shaped BECs at $3\le \lambda \le 4.5$ and (b) blood-cell-shaped BECs at $7\le \lambda \le 8$ at an inverse temperature of $\beta =1$. Depending on the trap aspect ratio, the decay rate shows a similar behavior as the activation energy in Fig. 4. The ranges of the trap aspect ratios and the scattering length are the same as in Fig. 4 and the range of the decay rate ($1\le \Gamma \le 3\times {10}^3$) corresponds to mean lifetimes $\tau ={\Gamma }^{-1}$ of $1.3\le \tau \le 3800\mathrm{ms}$ for a ${}^{52}$Cr BEC of $N=10000$ atoms. The inverse temperature $\beta =1$ corresponds to $T=20$nK.

Figure 6

Schematic description of the procedure to systematically access collectively excited stationary states which bifurcate from the ground state. Shown is the typical bifurcation scenario between the ground state and the excited states with $m$-fold rotational symmetry. The solid lines depict the states which cross the bifurcation smoothly and the dashed lines represent the emerging states. The numbers indicate the single steps of the procedure described in the text. Top: In an axisymmetric trap [$s=0$ in Eq. (2)], the ground state with $m=0$ passes the bifurcation smoothly and the $m\ne 0$ states bifurcate at a certain value $a_{\mathrm{bif}}$ of the $s$-wave scattering length. Center: The bifurcation scenario changes when the rotational symmetry of the external trap is changed ($s\ne 0$) in a way that the state which crosses the bifurcation smoothly has an $m\ne 0$ rotational symmetry. Bottom: The density profiles show the behavior of the wave function during steps 1–5 exemplarily for the $m=2$ bifurcation. The shape of the external trap is indicated by the contours.