Variational calculations for anisotropic solitons in dipolar Bose-Einstein condensates

We present variational calculations using a Gaussian trial function to calculate the ground state of the Gross-Pitaevskii equation (GPE) and to describe the dynamics of the quasi-two-dimensional solitons in dipolar Bose-Einstein condensates (BECs). Furthermore, we extend the ansatz to a linear superposition of Gaussians, improving the results for the ground state to exact agreement with numerical grid calculations using imaginary time and the split-operator method. We are able to give boundaries for the scattering length at which stable solitons may be observed in an experiment. By dynamic calculations with coupled Gaussians, we are able to describe the rather complex behavior of the thermally excited solitons. The discovery of dynamically stabilized solitons indicates the existence of such BECs at experimentally accessible temperatures.

Figure 1

Three-dimensional potential $V_h(\mathit{q})$ for ${\gamma }_y=2000$ visualized by isosurfaces. (a) Scattering length $a=0.1$. The ellipsoidal form of the isosurfaces marks the stable minimum, and the hyperbolic form of the isosurfaces close to zero marks the saddle point. In (b) the potential is rotated to show that the saddle point lies at smaller $q_y$ values than the minimum. (c) Same as (a) but at scattering length $a=0.08,$ lowered toward the bifurcation point. The minimum and the saddle point approach each other.

Figure 2

Mean-field energy (a), chemical potential (b), and width parameters of the trial function for the stable ground state on a logarithmic scale (c) as functions of the scattering length for different values of trap frequencies.

Figure 3

Real and imaginary parts of the eigenvalues of the Jacobi matrix $J$ as a function of the scattering length for the trap frequency ${\gamma }_y=2000$. The stable branch shows purely imaginary eigenvalues; the unstable one has nonvanishing real parts. The eigenvalues of the stable ${\Lambda }_s$ and unstable branch ${\Lambda }_u$ coincide at the bifurcation point. The inset shows a magnification of the rectangle.

Figure 4

Periodic trajectories in the potential $V_{2\mathrm{D},h}$ at different values of energy for ${\gamma }_y=2000$ and scattering length $a=0.1$. The red dashed line denotes the surface of section for the PSOSs. There are two periodic trajectories that belong to symmetric and antisymmetric oscillations of the soliton. The symmetric oscillations are hardly visible, because they lie on top of one another and are almost parallel to the surface of section.

Figure 5

PSOSs for the dynamics of the two-dimensional potential $V_{2\mathrm{D},h}$ in Eq. (16) in the rotated coordinates defined in Eq. (17), with trap frequency ${\gamma }_y=2000$ and (a) $a=0.1$, $E_{\mathrm{mf}}=1999.5$; (b) $a=0.08$, $E_{\mathrm{mf}}=1995$; (c) $a=0.08$, $E_{\mathrm{mf}}=1999$. In (a), the motion is completely regular. In (b), lowering the scattering length toward the bifurcation point causes chaotic dynamics to appear. In (c), increasing the mean-field energy close to $E_{\mathrm{mf}}={\gamma }_y$ at constant scattering length enlarges the chaotic regions in the Poincaré map.

Figure 6

Three-dimensional trajectory of a soliton at trap frequency ${\gamma }_y=2000$, scattering length $a=0.1$, and mean-field energy $E_{\mathrm{mf}}=3000,$ far above the threshold $E_{\mathrm{mf}}={\gamma }_y$, where dissolving is possible. (a,b) Time dependence of the expectation values $q_{\sigma }(t)$, where $\sigma =x,y,z$. (c) Lissajous-type motion of the projection in the $(q_x,q_z)$ plane.

Figure 7

ITE of ten coupled Gaussians. The parameters used are ${\gamma }_y=2000$, $a=0.1$. Plotted are, from upper to lower panels, the real part of the width parameters, the imaginary part of the width parameters, real and imaginary parts of $\gamma$, and the mean-field energy. The mean-field energy converges to the ground state of the system as described in the text. For comparison, the mean-field energy in a fast calculation with low integration accuracy is shown as well.

Figure 8

(a) Mean-field energy as a function of the scattering length of the ground state of a soliton at trap frequency ${\gamma }_y=2000,$ obtained by the ITE with different numbers of Gaussians. For comparison, the result for a single Gaussian obtained by a nonlinear root search and the results obtained by numerical grid calculations are plotted as well. (b) The chemical potential for the same parameters as in (a). (c) The temperature for different particle numbers belonging to the difference between upper-limit $E_{\mathrm{mf}}={\gamma }_y$ and ground-state energy.

Figure 9

Same as Fig. 8, but for trap frequency ${\gamma }_y=20\hspace{0.5em}000$. Compared to the results for ${\gamma }_y=2000$, the range of the scattering length is smaller but the temperature is higher.

Figure 10

Mean-field energy for trap frequency ${\gamma }_y=2000$ and fixed scattering length $a=0.1$. The mean-field energy converges fast with an increasing number of Gaussians.

Figure 11

Dynamics with two coupled Gaussians for trap frequency ${\gamma }_y=2000$, scattering length $a=0.1$, and mean-field energy $E_{\mathrm{mf}}=2100$. In (a) the parameters $q_x$ and $q_z$, and in (b) the parameter $q_y$, in the trap direction are shown as functions of time. (c) Projection of the trajectory in the $(q_x,q_z)$ plane. The soliton does not dissolve, although the mean-field energy is quite far above the threshold $E_{\mathrm{mf}}={\gamma }_y$.