Energetically stable singular vortex cores in an atomic spin-1 Bose-Einstein condensate

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Figure 1
Schematic illustration of two vortex-core structures with the same topology for a singly quantized singular vortex in the polar phase of a spin-1 condensate. In (a) the atom density vanishes at the vortex-line singularity with the core size determined by the characteristic length scale $ξ_{n}$ (healing length) associated with the spin-independent interaction strength. In (b) the atom density is nonvanishing in the core region, whose size is determined by the characteristic length scale $ξ_{F}$ of the spin-dependent interaction strength. The vortex-line singularity has now split into two half-quantum vortices with the atoms in the ferromagnetic phase at the precise location of the singularities. In both panels we show the nematic axis as dashed lines and the dotted line in (b) indicates a disclination plane for the nematic axis. Inside the core region (shaded area) of (b) the broken symmetry of the polar ground-state manifold is restored (as explained in the text). Outside the core the topological properties of the vortex are the same as those in (a).Reuse & Permissions
Figure 2
Schematic illustrations of FM vortex states. (a) The nonsingular, coreless vortex is formed as a combined disgyration of the spin vector (cones) and a spin rotation about the local spin vector (indicated by the orthogonal vectors). The vortex is nonsingular and the spin texture is continuous. (b) The singular FM spin vortex is formed as a radially oriented disgyration of the spin vector around the singular core, which is filled by the polar phase. (c) The class of singular vortices contains several different spin configurations, for example the cross-disgyration shown. They can all be deformed to the singular spin vortex through local spin rotations. (d) Energies of the stable coreless (lower line) and singular vortices as functions of the frequency of rotation in an isotropic trap using $N c_{0} = 1000 ℏ ω_{⊥} l_{⊥}^{3}$ and $N c_{2} = - 320 ℏ ω_{⊥} l_{⊥}^{3}$ . The coreless vortex is lower in energy in the whole frequency range.Reuse & Permissions
Figure 3
Energetic stability of the coreless (a) and the core-deformed singular vortex (b) in an isotropic trap for varying spin-dependent interaction strength $c_{2} < 0$ . The spin-independent interaction is fixed at $N c_{0} = 1000 ℏ ω_{⊥} l_{⊥}^{3}$ . Blue dots ( $•$ ) indicate that the vortex is energetically stable. A black plus ( $+$ ) indicates where the initial vortex leaves the cloud, whereas red crosses ( $\times$ ) mark where additional vortices nucleate due to rotation. A blue (black) vertical line marks $c_{0} / c_{2} ≃ - 216$ relevant for $^{87}$ Rb [65]. Note that with the parameters used here, this yields $N | c_{2} | = 4.6 ℏ ω_{⊥} l_{⊥}^{3}$ . The line thus falls very nearly on top of the vertical axis and the two cannot be distinguished in the figure.Reuse & Permissions
Figure 4
Numerically calculated spin textures in the stable FM vortex states in a rotating trap. The spin vector is shown in a cut perpendicular to the $z$ axis (the axis of rotation). (a) The spin vector in the coreless vortex exhibits a characteristic fountainlike structure and maintains $| 〈 \hat{F} 〉 | = 1$ everywhere. (b) In the relaxed singular vortex, the spin vector winds by $2 π$ around the $x$ axis on a path encircling the singular vortex core (indicated by the dot), in which $| 〈 \hat{F} 〉 | \to 0$ . This texture can be continuously deformed into that shown in Fig. 2b.Reuse & Permissions
Figure 5
Split core of the singular FM vortex and restoration of a single core with explicit axial symmetry of the spinor-component densities illustrated by isosurfaces of the spinor wave-function components $n | ζ_{+} |^{2}$ [red (medium gray)], $n | ζ_{0} |^{2}$ [green (light gray)], and $n | ζ_{-} |^{2}$ [blue (dark gray)]. (a) In the spinor basis along the $z$ axis, the vortex lines in $ζ_{+}^{(z)}$ , $ζ_{0}^{(z)}$ , and $ζ_{-}^{(z)}$ separate and the atom density is nonzero everywhere. (b) The axial symmetry of the density in each spinor component is restored by transforming the spinor to the basis of spin projection onto the $x$ axis. Vortex lines with opposite circulation in $ζ_{\pm}^{(x)}$ overlap. $ζ_{0}^{(x)}$ (not shown) does not exhibit any vortex line [cf. Fig. 6b]. See also the Appendix for a qualitative analytic discussion of the relation between $ζ^{(x)}$ and $ζ^{(z)}$ .Reuse & Permissions
Figure 6
(a) Densities in the three spinor components $ζ_{+}^{(z)}$ (red line marked by $+$ ), $ζ_{0}^{(z)}$ (green line marked by 0), and $ζ_{-}^{(z)}$ (blue line marked by $-$ ) on the axis connecting the vortex lines in the spinor components [cf. Fig. 5a]. (b) Densities in $ζ_{+}^{(x)}$ (red line marked by $+$ ), $ζ_{0}^{(x)}$ (green line marked by 0), and $ζ_{-}^{(x)}$ (blue line marked by $-$ ) on the same spatial axis after spinor-basis transformation. $| 〈 \hat{F} 〉 |$ (black dash-dotted line) goes to zero in the vortex core (the apparent nonzero minimum is due to the finite numerical resolution) which is filled by $ζ_{0}^{(x)}$ , keeping the density nonzero everywhere.Reuse & Permissions
Figure 7
Axially symmetric spin vortex. (a) Densities of the spinor components together with the spin magnitude along a radial cut [lines and labels as in Fig. 6a]. The vortex lines in $ζ_{\pm}$ overlap perfectly at the position of the vortex core. (b) Relaxed spin profile in the $x y$ plane, showing the characteristic radial disgyration of the spin vector around the singular core. At large radii the spin vector bends out of the $x y$ plane. The vortex-line singularity is marked by a dot at the center.Reuse & Permissions
Figure 8
Spin magnitude $| 〈 \hat{F} 〉 |$ , showing the core of the stable singular vortex of Fig. 6 in the FM phase, shown in the $x y$ plane (a) and $y z$ plane (b). Outside the vortex core, $| 〈 \hat{F} 〉 | = 1$ [dark red (dark gray)] in the FM order-parameter manifold. In the core, the singularity is accommodated by enforcing $| 〈 \hat{F} 〉 | = 0$ (white) while maintaining nonzero density. The size of the core region is determined by the spin healing length $ξ_{F}$ .Reuse & Permissions
Figure 9
Stability of the FM coreless (a) and singular (b) vortices in a highly oblate trap, $ω_{⊥} / ω_{z} = 0.1$ , with $N c_{0} = 50 ℏ ω_{⊥} l_{⊥}^{3}$ and $- 48 ℏ ω_{⊥} l_{⊥}^{3} ⩽ N c_{2} ⩽ - 5 ℏ ω_{⊥} l_{⊥}^{3}$ . (Symbols as in Fig. 3.)Reuse & Permissions
Figure 10
Effects of linear and quadratic Zeeman splitting on the stability of the FM vortices in an oblate trap ( $ω_{⊥} / ω_{z} = 0.1$ ). In all panels $N c_{0} = 50 ℏ ω_{⊥} l_{⊥}^{3}$ and $N c_{2} = - 10 ℏ ω_{⊥} l_{⊥}^{3}$ . (a) Stability of a coreless vortex in the presence of linear Zeeman splitting. The vortex state becomes unstable for $g_{1} | B | ≳ 0.2 ℏ ω_{⊥}$ . (b) The singular vortex remains stable despite linear Zeeman splitting. (c) Stability of the coreless vortex in the presence of quadratic Zeeman splitting. A stable region is found at all investigated values of $g_{2} | B |$ . (d) Stability of the singular vortex in the presence of quadratic Zeeman splitting. The vortex becomes unstable for a relatively small negative $g_{2} | B |$ . (Symbols as in Fig. 3.)Reuse & Permissions
Figure 11
Stable core structure of the singular vortex in the polar phase shown in the $x y$ plane. (a),(b) Densities in $ζ_{+}^{(z)}$ and $ζ_{0}^{(z)}$ , respectively. (c),(d) The corresponding phases. ( $ζ_{-}^{(z)}$ is identical to $ζ_{+}^{(z)}$ up to a global $π$ phase shift.) The spinor wave function exhibits vortex lines with highly deformed anisotropic cores in the spinor components.Reuse & Permissions
Figure 12
Splitting of the singly quantized vortex into two half-quantum vortices. (a) Spin magnitude $| 〈 \hat{F} 〉 |$ [color map from white ( $| 〈 \hat{F} 〉 | = 0$ ) to red (dark gray) ( $| 〈 \hat{F} 〉 | = 1$ )] together with the spin vector (arrows), showing the FM cores with nonvanishing density. The spins are antiparallel in the two cores. (b) Nematic axis $\hat{d} (r)$ together with the vortex cores [indicated by green (light gray) isosurfaces of $| 〈 \hat{F} 〉 |$ , with increasing spin magnitude indicated by the color gradient inside]. Away from the vortex cores the topology of the initial singly quantized vortex is preserved. In the core region, $\hat{d}$ winds by $π$ about each half-quantum vortex core. For visualization purposes, the unoriented $\hat{d}$ field is shown as cones. Here a quadratic Zeeman shift has been introduced to ensure that $\hat{d}$ lies in the $x y$ plane and the spins align with the $z$ axis.Reuse & Permissions
Figure 13
Spinor wave function of the stable singular vortex state from Fig. 11 after spinor-basis transformation such that spin is quantized along the $y$ axis. (a),(b) Densities in $ζ_{+}^{(y)}$ and $ζ_{-}^{(y)}$ , respectively. (c),(d) The corresponding phases. The component $ζ_{0}^{(y)}$ (not shown) is unpopulated. The previously complex structure can now be identified as a pair of half-quantum vortices.Reuse & Permissions
Figure 14
(a) Density profiles of the spinor components on the axis connecting their density minima (cf. Fig. 11). Lines are labeled with the spinor-component index. Note that $| ζ_{\pm} |$ exactly overlap. (b) Spinor-component density profiles in $ζ^{(y)}$ after basis transformation (cf. Fig. 11) plotted along the axis connecting the half-quantum vortices. The spin magnitude $| 〈 \hat{F} 〉 |$ (black dash-dotted line) shows the FM cores. Unpopulated $ζ_{0}^{(y)}$ is not shown.Reuse & Permissions
Figure 15
Energetic stability of the split-core singular vortex in the polar phase. (a) Stability in the isotropic trap for varying $c_{2}$ using $N c_{0} = 1000 ℏ ω_{⊥} l_{⊥}^{3}$ . The vertical line marks the value $c_{0} / c_{2} ≃ 31$ for $^{23}$ Na [71]. (b) Stability in the oblate trap ( $ω_{⊥} / ω_{z} = 0.1$ ) for varying $c_{2}$ using $N c_{0} = 50 ℏ ω_{⊥} l_{⊥}^{3}$ . (c),(d) Stability of the split-core singular vortex in the presence of linear and quadratic Zeeman splitting, respectively. Increasing linear Zeeman splitting renders the singular vortex unstable, whereas the stability is robust against quadratic Zeeman splitting. In the slowly rotating region for all panels, the instability of the split-core singular vortex may be either towards the vortex-free state, or towards the state with a single half-quantum vortex. (Symbols as in Fig. 3.)Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum information

Energetically stable singular vortex cores in an atomic spin-1 Bose-Einstein condensate

Justin Lovegrove, Magnus O. Borgh, and Janne Ruostekoski

Phys. Rev. A 86, 013613 – Published 12 July 2012; Erratum Phys. Rev. A 89, 039902 (2014)

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Erratum: Energetically stable singular vortex cores in an atomic spin-1 Bose-Einstein condensate [Phys. Rev. A 86, 013613 (2012)]

Justin Lovegrove, Magnus O. Borgh, and Janne Ruostekoski

Phys. Rev. A 89, 039902 (2014)

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Physical Review A

covering atomic, molecular, and optical physics and quantum information

Energetically stable singular vortex cores in an atomic spin-1 Bose-Einstein condensate

Justin Lovegrove, Magnus O. Borgh, and Janne Ruostekoski

Phys. Rev. A 86, 013613 – Published 12 July 2012; Erratum Phys. Rev. A 89, 039902 (2014)

Abstract

Erratum

Erratum: Energetically stable singular vortex cores in an atomic spin-1 Bose-Einstein condensate [Phys. Rev. A 86, 013613 (2012)]

Justin Lovegrove, Magnus O. Borgh, and Janne Ruostekoski

Phys. Rev. A 89, 039902 (2014)

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