Stability and internal structure of vortices in spin-1 Bose-Einstein condensates with conserved magnetization

We demonstrate how conservation of longitudinal magnetization can have pronounced effects on both stability and structure of vortices in the atomic spin-1 Bose-Einstein condensate by providing a systematic characterization of nonsingular and singular vortex states. Constructing spinor wave functions for vortex states that continuously connect ferromagnetic and polar phases, we systematically derive analytic models for nonrotating cores of different singular vortices and for composite defect states with distinct small- and large-distance topology. We explain how the conservation law provides a stabilizing mechanism when the coreless vortex imprinted on the condensate relaxes in the polar regime of interatomic interactions. The resulting structure forms a composite defect: The inner ferromagnetic coreless vortex deforms toward an outer singly quantized polar vortex. We also numerically show how other even more complex hierarchies of vortex-core topologies may be stabilized. Moreover, we analyze the structure of the coreless vortex also in a ferromagnetic condensate and show how reducing magnetization leads to a displacement of the vortex from the trap center and eventually to the deformation and splitting of its core where a singular vortex becomes a lower-energy state. For the case of singular vortices, we find that the stability and the core structure are notably less influenced by the conservation of magnetization.

Figure 1

Numerically calculated spin textures in the $x-y$ plane for the coreless vortex in the FM regime with an initial magnetization $M=0.4$ (a) not conserved and (b) conserved throughout the relaxation process. (c) The same for a conserved magnetization of $M=0.2$, showing a displacement of the coreless vortex relative to the more strongly magnetized case. (d) Displacement and (e) angular momentum of the coreless vortex in a trap rotating at $\mathrm{\Omega }/\omega =0.3$ for different values of the conserved magnetization (black dots) compared with the angular momentum of an axisymmetric coreless vortex at the center of the trap, for the same magnetization (blue line). (f) Numerically calculated magnetization of the energy-minimizing coreless vortex in the FM regime as a function of rotation frequency, where magnetization-conserving relaxation has not been enforced.

Figure 2

Energetic stability of the coreless vortex in the polar (left) and FM (right) interaction regimes; ($\blacksquare )$ stable coreless vortex; ($\square )$ stable effective two-component coreless vortex; ($▵)$ instability towards a half-quantum vortex; ($\circ )$ pair of half-quantum vortices (polar regime) or singular vortex (FM regime); ($+)$ vortex-free state; ($\times )$ nucleation of additional vortices.

Figure 3

Coreless vortex in strongly magnetized condensate. (Left) Spin magnitude (color scale) and spin vector (arrows). Note how the spin magnitude falls below 1 away from the vortex center. (Right) Spinor wave function along the $y$ axis, showing how the strong magnetization leads to an effective two-component regime, where ${\zeta }_{-}$ is depleted and ${\zeta }_0$ exhibits a $2\pi$ phase winding.

Figure 4

(Left) Spin profile $\langle \widehat{\mathbf{F}}\rangle$ (arrows) and $|\langle \widehat{\mathbf{F}}\rangle |$ (color gradient and arrow lengths) of the coreless vortex in the polar regime, interpolating between FM and polar phases and displaying the characteristic fountain texture inside the core of a singular polar vortex. (Right) The corresponding superfluid velocity $\mathbf{v}$ and its magnitude (arrows) and circulation density $\mathcal{V}=\rho \mathbf{v}·\widehat{\mathbf{\phi }}$ (color gradient) continuously interpolating from nonsingular to singular circulation.

Figure 5

Stable nematic coreless vortex in a BEC with polar interactions. (Left) The unoriented $\widehat{\mathbf{d}}$ vector (cylinders) exhibits the coreless fountainlike texture. The circulation density $\mathcal{V}=\rho \mathbf{v}·\widehat{\mathbf{\phi }}$ (color gradient) shows the composite-vortex structure, interpolating between the noncirculating polar phase to the outer singly quantized FM vortex. (Right) Corresponding spin texture $\langle \widehat{\mathbf{F}}\rangle$ (arrows) and spin magnitude $|\langle \widehat{\mathbf{F}}\rangle |$ (color gradient and arrow lengths), showing the core region. Conservation of magnetization forces the BEC into the FM phase away from the vortex line.

Figure 6

Numerically calculated spin vector (arrows) and magnitudes (color gradient) for the singular vortex of the FM phase in the $x-y$ plane with a conserved magnetization of (a) $M=0.2$ and (b) $M=0.5$. Arrow lengths scale with the spin magnitude. The core structure remains as described in Ref. [26], with the conserved magnetization leading to a local rotation in spin space as the vortex core deforms.

Figure 7

Spin vector (arrows) and magnitude (color gradient) profiles in the $x-z$ plane for a half-quantum vortex with an initial magnetization $M=0.2$ which is (a) not conserved and (b) conserved. Arrow lengths scale with spin magnitude. The magnetization arises from the outer regions, not the FM core.

Figure 8

Numerically calculated spin magnitude (color gradient) and vector (arrows) profiles of the singular polar vortex in the $x-y$ plane when an initial magnetization $M=0.2$ is (a) not conserved and (b) conserved. Arrow lengths scale with spin magnitude. The singular vortex deforms to a pair of half-quantum vortices with FM cores in both cases.

Figure 9

(a) Numerically calculated spin texture (arrows) and magnitude (color gradient) in the $x-y$ plane for the composite vortex in the FM regime with $M=0.6$. Arrow lengths scale with spin magnitude. (b) Nematic axis profile (cylinders) and circulation (color gradient) for the same, showing the noncirculating inner FM core and winding of $\widehat{\mathbf{d}}$ by $\pi$.

Figure 10

Spin magnitude (color scale) and spin vector (arrows) showing composite vortex states arising as a singly quantized vortex in a FM condensate relaxes in the presence of an engineered quadratic Zeeman shift with Laguerre-Gaussian profile that forces the condensate into the polar phase in an annulus surrounding the vortex core. (Left) Trap rotation $\mathrm{\Omega }=0.125\omega$. A coreless vortex appears in the inner FM core, surrounded by a singly quantized polar vortex, together forming the extended core of a singular FM vortex. (Right) $\mathrm{\Omega }=0.135\omega$. Rotation causes a multiply quantized vortex to form in the outer FM region.