Quasi-one- and quasi-two-dimensional symbiotic solitons bound by dipolar interaction

We study the formation of quasi-one- (quasi-1D) and quasi-two-dimensional (quasi-2D) symbiotic solitons bound by an interspecies dipolar interaction in a binary dipolar Bose-Einstein condensate. These binary solitons have a repulsive intraspecies contact interaction stronger than the intraspecies dipolar interaction, so that they can not be bound in isolation in the absence of an interspecies dipolar interaction. These symbiotic solitons are bound in the presence of an interspecies dipolar interaction and zero interspecies contact interaction. The quasi-1D solitons are free to move along the polarization $z$ direction of the dipolar atoms, whereas the quasi-2D solitons move in the $x$-$z$ plane. To illustrate these, we consider a $^{164}$Er-$^{166}$Er mixture with scattering lengths $a$($^{164}$Er)$=81a_0$ and $a$($^{166}$Er)$=68a_0$ and with dipolar lengths $a_{\mathrm{dd}}$($^{164}$Er)$\approx a_{\mathrm{dd}}$($^{166}$Er)$\approx 65a_0$, where $a_0$ is the Bohr radius. In each of the two components $a> a_{\mathrm{dd}}$, which stops the binding of solitons in each component in isolation, whereas a binary quasi-1D or a quasi-2D $^{164}$Er-$^{166}$Er soliton is bound in the presence of an interspecies dipolar interaction. The stationary states were obtained by imaginary-time propagation of the underlying mean-field model; dynamical stability of the solitons was established by real-time propagation over a long period of time.