Variational calculations on multilayer stacks of dipolar Bose-Einstein condensates

We investigate a multilayer stack of dipolar Bose-Einstein condensates in terms of a simple Gaussian variational ansatz and demonstrate that this arrangement is characterized by the existence of several stationary states. Using a Hamiltonian picture we show that in an excited stack there is a coupled motion of the individual condensates by which they exchange energy. We find that for high excitations the interaction between the single condensates can induce the collapse of one of them. We furthermore demonstrate that one collapse in the stack can force other collapses, too. We discuss the possibility of experimentally observing the coupled motion and the relevance of the variational results found to full numerical investigations.

Figure 1

A multilayer stack of dipolar BECs, each placed in a trap that is assumed to be very oblate. The single condensates are displaced by a distance $\Delta$, respectively, and coupled by the long-range dipole-dipole interaction (cf. [6]).

Figure 2

Mean-field energy of the stationary states for a stack of $N_s=3$ condensates for a distance $\Delta =0.07$, scaled trap frequency $N^2\mathrm{\gamma ̄}=1300$, and aspect ratio $\lambda =340$ with $N_b=6$ different branches. (Inset) The different branches arise in tangent bifurcations at different values of the scattering length. The symbols “s” and “u” denote weather of not the individual BECs are stable or unstable.

Figure 3

Eigenvalues of the Jacobian matrix $J$ for a stack of $N_s=3$ BECs and different arrangements in dependence on the scattering length. The physical parameters are the same used for the calculation of the mean-field energy in Fig. 2. The state “s-s-s” (solid red lines) is the only one with exclusively negative eigenvalues, and, in general, the eigenvalues corresponding to a motion in $z$ direction (a) differ by several orders of magnitude from those corresponding to the motion in radial direction (b).

Figure 4

Time evolution of a two-layer stack of dipolar BECs described by a particle moving in the external potential (7) for $\Delta =0.035$ and a scattering length of $a/a_d=-0.1$. At $t=0$, the particle is placed in the minimum of the potential $V$ with nonzero initial momentum $p_{\rho 1}\ne 0$. The two condensates represented by the particle exchange their excitation energy periodically. For a small initial momentum $p_{\rho 1}$ (a), the energy exchange happens in a shorter period of time than it does for large values of the initial momentum (b).

Figure 5

Normal modes of the coupled radial motion of two interacting BECs. The two BECs either oscillate in phase (on the left) or with a phase shift of $\pi$ (on the right) and different oscillation frequencies corresponding to the different normal modes.

Figure 6

The frequency $\mathrm{\omega ̃}=|{\omega }_1-{\omega }_2|/2$ that can be interpreted as a characteristic frequency for the energy exchange between two condensates of Fig. 4. The frequency depends crucially on the distance $\Delta$ between the condensates and the scattering length $a/a_d$ and reaches its highest value for small distances $\Delta$ and $a\to a_{\mathrm{cr}}$.

Figure 7

Frequency spectra of the radial oscillation of the single condensates in the stack of two BECs of Fig. 4 for different excitation energies $E^{*}$. The single peaks indicate the fundamental frequencies, and the peak heights are plotted in arbitrary units.

Figure 8

Time evolution of the radial extension of the two BECs of Fig. 4 described by a particle moving in the external potential $V$ with initial momenta $p_{\rho j},p_{zj}\ne 0$ and initial values for the generalized coordinates that are not the fixed points of the Hamiltonian equations of motion. The mean-field energy of the system lies slightly above the saddle point of the external potential. (a) The extension of the BECs oscillates and they exchange their excitation energy. (b) The last few oscillations. (c) The second BEC ($j=2$) has transferred its whole kinetic energy to the first one which is now highly excited and whose extension finally reaches $q_{\rho 1}=0$, meaning the collapse of the condensate.

Figure 9

Time evolution of a stack of $N_s=3$ BECs. The multilayer stack of dipolar BECs is assumed to be in the ground state at a scattering length that lies slightly above the critical value and does, therefore, not change in time (dashed lines). Reducing the scattering length below the critical value, all coordinates $q_{\rho j}$ begin to shrink, representing the contraction of the BECs. (a) The central BEC (dashed-dotted line) collapses first but at a distance of $\Delta =0.035$; this also causes the collapse of the other BECs (dotted line). (b) The same situation with a distance of $\Delta =0.07$. The central condensate again collapses, but the coupling is not strong enough to determine the other BECs to collapse.