Stable, mobile, dark-in-bright, dipolar Bose-Einstein-condensate solitons

We demonstrate robust, stable, mobile, quasi-one-dimensional, dark-in-bright dipolar Bose-Einstein-condensate (BEC) solitons with a notch in the central plane formed due to dipolar interaction for repulsive contact interaction. At medium velocity the head-on collision of two such solitons is found to be quasielastic with practically no deformation. A proposal for creating dipolar dark-in-bright solitons in laboratories by phase imprinting is also discussed. A rich variety of such solitons can be formed in dipolar binary BECs, where one can have a dark-in-bright soliton coupled to a bright soliton or two coupled dark-in-bright solitons. The findings are illustrated using numerical simulation in three spatial dimensions by employing realistic interaction parameters for a dipolar ${}^{164}$Dy BEC and a binary ${}^{164}$Dy-${}^{162}$Dy BEC.

Figure 1

Stability phase diagram showing the critical number of atoms $N_{\mathrm{crit}}$ in a quasi-1D dipolar dark-in-bright (DIB) and bright ($b$) BEC soliton of ${}^{164}$Dy or ${}^{168}$Er atoms from numerical calculation. The system is repulsive and unbound for $a\gtrsim a_{\mathrm{dd}}$. Stable quasi-1D solitons appear for $a\lesssim a_{\mathrm{dd}}$ and the number of atoms $N$ below the critical number $N_{\mathrm{crit}}$. The oscillator length $l=1$ $\mu$m.

Figure 2

The 3D isodensity contour (${\left|\varphi \right|}^2$) of (a) a dark-in-bright and (b) a bright soliton of 1000 ${}^{164}$Dy atoms with $a=80a_0$. The dimensionless lengths $x$, $y,$ and $z$ are in units of $l$ $(\equiv 1$ $\mu$m). The density on the contour is ${10}^7$ atoms/cm${}^3$ compared to the central density in the bright soliton of (b) of about ${10}^{12}$ atoms/cm${}^3$.

Figure 3

Linear axial density of (a) two colliding dark-in-bright solitons of 1000 ${}^{164}$Dy atoms each from Fig. 2 and (b) the colliding dark-in-bright soliton of Fig. 2 and a bright soliton of 500 ${}^{164}$Dy atoms with $a=80a_0$.

Figure 4

(a) Linear axial density of the dipolar dark-in-bright soliton of 1000 ${}^{164}$Dy atoms with $a=80a_0$ and $l=1$ $\mu$m upon real-time propagation. The initial state has been modified to have the central zero of the dark-in-bright soliton at $z/l\approx -2$. (b) Initial ($t=0$) and final ($t/t_0=100$) profiles of linear axial densities.

Figure 5

Creating a dark-in-bright soliton in a real-time simulation of a phase-imprinted bright soliton of 1000 ${}^{164}$Dy atoms with $a=80a_0$ and $l=1$ $\mu$m. The linear axial density of the bright soliton (blue) is shown for (a) small ($5>t/t_0>0$) and (b) large ($20>t/t_0>16$) times. (c) The constant density of the stationary dark-in-bright soliton (red) of 1000 ${}^{164}$Dy atoms obtained by an imaginary-time routine with $a=80a_0$ and $l=1$ $\mu$m is also shown for comparison.

Figure 6

Stability phase diagram for the critical number of ${}^{162}$Dy atoms in a binary ${}^{164}$Dy-${}^{162}$Dy soliton with 1000 ${}^{164}$Dy atoms versus the interspecies scattering length with (a) two coupled bright ($b$-$b$) solitons, (b) a dark-in-bright (DIB) ${}^{164}$Dy soliton coupled to a bright (DIB-$b$) ${}^{162}$Dy soliton, and (c) two coupled DIB (DIB-DIB) solitons. The parameters are $a{(}^{164}$Dy) = $a{(}^{162}$Dy) = $120a_0$ and $l=1$ $\mu$m.

Figure 7

The 3D isodensity contour $\left(\right|{\varphi }_i\left(\mathbf{r}\right){|}^2)$ of two dark-in-bright solitons in the binary ${}^{164}$Dy-${}^{162}$Dy mixture: (a) ${}^{164}$Dy and (b) ${}^{162}$Dy profiles. Also shown is the same of a dark-in-bright and a bright soliton in the binary ${}^{164}$Dy-${}^{162}$Dy mixture: (c) ${}^{164}$Dy and (d) ${}^{162}$Dy profiles. The parameters are $a{(}^{164}{\text{Dy-}}^{162}\text{Dy})=100a_0$, $a{(}^{164}\text{Dy})=a{(}^{162}\text{Dy})=120a_0$, $l=1$ $\mu$m, $N{(}^{164}\text{Dy})=1000$, and $N{(}^{162}\text{Dy})=3000$. The density on the contour is ${10}^7$ atoms/cm${}^3$.