Creating two-dimensional bright solitons in dipolar Bose-Einstein condensates

We propose a realistic experimental setup for creating quasi-two-dimensional (2D) bright solitons in dipolar Bose-Einstein condensates (BECs), the existence of which was proposed by Tikhonenkov et al. [Phys. Rev. Lett. 100, 090406 (2008)]. A challenging feature of the expected solitons is their strong inherent anisotropy due to the necessary in-plane orientation of the local moments in the dipolar gas. This may be the first chance of making multidimensional matter-wave solitons, as well as solitons featuring the anistropy due to their intrinsic dynamics. Our analysis is based on the extended Gross-Pitaevskii equation, which includes three-body losses and noise in the scattering length, induced by fluctuations of currents inducing the necessary magnetic fields, which are factors crucial to the adequate description of experimental conditions. By means of systematic three-dimensional (3D) simulations, we find a ramping scenario for the change of the scattering length and trap frequencies which results in the creation of robust solitons, that readily withstand the concomitant excitation of the condensate.

Figure 1

Ramping of (a) scattering length $a$ and (b) trap frequencies. Only the first 100 ms are shown. In the course of the remaining time of the simulations, the scattering length is held at a constant value, while the trap has already been switched off. In the experiment, the scattering length cannot be held at an exact value, but contains some noise, which is included into our scheme, as shown by the green line in (a).

Figure 2

The formation of a soliton. Left column: The scattering length is ramped without the noise. Right column: The random noise with the root-mean-square value of $\Delta a=0.5a_{{\mathrm{B}}^{}}$ was added to the scattering length. Shown are absorption images of the condensate density, with ${\left|\Psi \right|}^2$ integrated along the $y$ axis and normalized to the maximum value. In both cases, the soliton is being shaped in the first two frames, and is still expanding at $t=217$ ms. Without the noise, the condensate self-focuses already at $t=303$ ms, while this is slowed down when the noise is included. In both cases, the refocusing occurs at the evolution stage when the external trap has already been switched off completely, hence it is the DDIs which are responsible for the self-trapping. This is a clear indication of the existence of a stable excited soliton. It is important to note that the noise does not destroy the soliton, which is crucial for the experiments. The field of the view is $232\mu$m $\times 213\mu$m.

Figure 3

Root-mean-square extensions of the soliton in (a) $x$ and (b) $z$ directions for various values of the noise. Naturally the largest extension of the condensate increases when the noise is added to the scattering length. At the end of the simulations performed without the noise, the soliton has already started a second cycle of the oscillations.

Figure 4

The number of atoms in the condensate in the course of the formation of the soliton. Most atoms are lost during the first 50 ms, when the density is still large and the soliton has not yet been formed completely. When the soliton starts its first cycle of oscillations, the loss rate rapidly drops, and at the end of the simulation, the condensate still keeps more than $80\%$ of the initial atoms. For the noise amplitude of $1a_{{\mathrm{B}}^{}}$, the losses are not visible at all after the trap has been switched off.