Stable Solitons in Three Dimensional Free Space without the Ground State: Self-Trapped Bose-Einstein Condensates with Spin-Orbit Coupling

By means of variational methods and systematic numerical analysis, we demonstrate the existence of metastable solitons in three dimensional (3D) free space, in the context of binary atomic condensates combining contact self-attraction and spin-orbit coupling, which can be engineered by available experimental techniques. Depending on the relative strength of the intra- and intercomponent attraction, the stable solitons feature a semivortex or mixed-mode structure. In spite of the fact that the local cubic self-attraction gives rise to the supercritical collapse in 3D, and hence the setting produces no true ground state, the solitons are stable against small perturbations, motion, and collisions.

Figure 1

$E_{\text{tot}}$ as a function of condensate’s size $L$, as per Eq. (2). The red solid, blue dashed, and green dot-dashed lines represent the energy’s variation when $\lambda =0$, and $\lambda >0$ does or does not satisfy condition (3), respectively.

Figure 2

Three-dimensional stable solitons are predicted by the variational calculation in blue shaded regions of the respective parameter planes. In (a), these are SVs at $\eta <1$ and MMs at $\eta >1$, with the boundary between them depicted by the black solid line. In (b), the entire stability area is filled by the solitons of both types, as the SVs and MMs have equal energies at $\eta =1$. The predictions are accurately confirmed by full numerical simulations, as indicated by red crosses and black dots, which indicate, respectively, the absence and presence of stable solitons for respective sets of parameters.

Figure 3

(a) Energies of the SVs and MMs, as predicted by the variational approach for $g=\lambda =1$ and $N=8$. The two curves cross at $\eta =1$, where the SV and MM have equal energies. (b) The numerically (blue circles) and variationally (red squares) found chemical potential vs the norm for the SVs at $g=\lambda =1$ and $\eta =0.3$. The numerical branch extends up to $N=N_{\max }$, in agreement with Eq. (9).

Figure 4

Density profiles of 3D solitons for $N=8$ and $g=\lambda =1$. (a) A SV for $\eta =0.3$, whose fundamental and vortical components $|{\psi }_{+}|$ and $|{\psi }_{-}|$ are plotted in (a1) and (a2), respectively. (b) A MM for $\eta =1.5$, with (b1) and (b2) displaying $|{\psi }_{+}|$ and $|{\psi }_{-}|$, respectively. In each subplot, different colors represent constant-magnitude surfaces, $|{\psi }_{\pm }|=(0.96,0.4,0.04)\times |{\psi }_{\pm }{|}_{\max }$.

Figure 5

The ratio of the spin populations as a function of velocity $v_z$ for the moving SV with $N=8$, $g=\lambda =1$, $\eta =0.3$, and $N_{\pm }\equiv \int d^{3}r|{\psi }_{\pm }(\mathbf{r}){|}^2$. The red dashed lines with squares are variational results, while the blue solid lines with circles are obtained numerically, using the imaginary-time integration in the moving reference frame.

Figure 6

Collisions of stable 3D SVs in the harmonic trap for $N=8$, $g=\lambda =1$, and $\eta =0.3$. Panels [(a1), (a2)], [(a3), (a4)], [(a5), (a6)], [(a7), (a8)], and [(a9), (a10)] display density distributions for $\mathrm{\Omega }=0.5$ at $t=0$, 1.2, 3.2, 4, and 6, respectively. Panels [(b1), (b2)], [(b3), (b4)], [(b5), (b6)], [(b7), (b8)], and [(b9), (b10)] display the distributions for $\mathrm{\Omega }=1$ at $t=0$, 1, 1.4, 2, and 3, respectively. In all panels, the left and right subplots display, severally, $|{\psi }_{+}|$ and $|{\psi }_{-}|$.