Abstract
Persistent topological defects and textures are particularly dramatic consequences of superfluidity. Among the most fascinating examples are the singular vortices arising from the rotational symmetry group SO(3), with surprising topological properties illustrated by Dirac’s famous belt trick. Despite considerable interest, controlled preparation and detailed study of vortex lines with complex internal structure in fully threedimensional spinor systems remains an outstanding experimental challenge. Here, we propose and implement a reproducible and controllable method for creating and detecting a singular SO(3) line vortex from the decay of a nonsingular spin texture in a ferromagnetic spin1 Bose–Einstein condensate. Our experiment explicitly demonstrates the SO(3) character and the unique spinor properties of the defect. Although the vortex is singular, its core fills with atoms in the topologically distinct polar magnetic phase. The resulting stable, coherent topological interface has analogues in systems ranging from condensed matter to cosmology and string theory.
Introduction
Quantised vortices are topological defects with universal properties that span seemingly disparate areas of science, such as highenergy physics, superconductivity, liquid crystals, and superfluids^{1}. Superfluids with internal degrees of freedom such as liquid ^{3}He (ref. ^{2}) and dilutegas spinor Bose–Einstein condensates^{3,4} (BECs) may exist in diverse stable phases, characterised by different broken symmetries of the full interaction Hamiltonian. Distinct states within a given phase of matter transform into one another in several ways, such as through rotations of spin and condensate phase. As a result, a rich phenomenology emerges consisting of topological defects and textures that resemble those predicted to exist in quantum field theories and cosmology^{1}.
In ordinary scalar superfluids, such as superfluid liquid ^{4}He and dilute BECs with frozen internal degrees of freedom, quantised vortices are characterised by the winding of the phase of the macroscopic wave function about any closed loop encircling the vortex line^{5,6}. The whole spectrum of phase values converges to the singular vortex line, at which the superfluid density vanishes.
In contrast, spin1 condensates are described by a threecomponent spinor, which admits both polar (P) and ferromagnetic (FM) groundstate magnetic phases. For atoms in the FM phase, the magnitude of the spin assumes its maximum value of one^{3,4}, and all of the different physical states are related to each other by spatial rotations of the spinor. The distinguishable states of the system are fully represented by the orientation of a local, orthonormal vector triad defined by the orientation of the atomic spin and rotations about it, corresponding to the elements of the group SO(3) of threedimensional (3D) spatial rotations. Mathematical analysis^{7} of this symmetry group indicates that vorticity must be carried either by coreless, nonsingular spin textures, or by quantised, singular vortices.
Quantised singular SO(3) vortices with even winding numbers have the unusual property that they are topologically equivalent to the defectfree state. When the local orientation of the vector triad describing the SO(3) vortex undergoes an even number of 2π rotations about an axis passing through the system, the triads can be locally and continuously reoriented, smoothly returning the system to a uniform configuration. Equivalently, joining two vortices with 2π winding each can cancel their net vorticity, either when they circulate oppositely or—less obviously—when they wind in the same sense. This essential property has been attributed to Dirac as his eponymous belt trick, a demonstration in which two 2π twists of a belt in the same direction may return it to its original configuration^{8}; but the concept also makes an appearance in diverse artistic contexts such as the Balinese candle dance. The significance of the belt trick to our work is that vortices with an odd number of 2π rotations of the vector triad are all equivalent to one another but not to the defectfree state.
In light of their peculiar properties, which have no correspondence in scalar quantum fluids, singular SO(3) vortices have attracted considerable attention in several different contexts. They have previously been described and indirectly detected in the superfluid liquid ^{3}HeA phase^{2,9,10}, where their direct visualisation is challenging. In spin1 BECs, they have been studied theoretically as the unique class of singular vortices in the FM phase^{11,12,13,14}. Of particular significance is the fact that, although the superfluid density in the FM phase must vanish along the line where the triad orientation is illdefined, the singular vortex can lower its energy by developing a superfluid core consisting of atoms in the spinless P phase that are excited out of the FM groundstate manifold^{12,14,15}. This phenomenon has been observed experimentally in the spontaneous vorticity of randomly appearing singular SO(3) defects in quasitwodimensional (2D) condensates during a rapid nonequilibrium phase transition^{16}, where the filled vortex cores were detected indirectly by their lack of longitudinal magnetisation. More recently, atomic condensates subjected to momentumdependent artificial gauge potentials were shown to support filledcore vortices^{17} closely related to those studied in our work. Related but topologically different halfquantum vortices have also been observed in the P phase in a quasi2D BEC^{18}. Despite these efforts, the controlled creation of singular SO(3) vortices remains an experimental challenge.
Here, in a striking manifestation of the topological constraints of the SO(3) order parameter, we transform a nonsingular vortex that is topologically equivalent to one with a 4π winding of the FM order parameter into a pair of spatiallyseparated singular SO(3) vortices with 2π winding each (Fig. 1). We thereby circumvent the smooth topological unwinding permitted by Dirac’s belt trick, dividing the equivalent of a 4π rotation into two 2π rotations that, once separated, cannot individually unwind. We find experimentally that the singular FM vortex cores are filled and expanded by atoms in the P phase. This establishes the existence of a coherent topological interface^{14,19}, where the order parameter continuously interpolates between the two magnetic phases within the vortex core. Such topological interfaces are universal across many areas of physics, including superfluid liquid ^{3}He at the boundary between coexisting A and B phases^{20,21}, earlyuniverse cosmology and superstring theory as domain walls^{22} and branes^{23}, and solidstate physics supporting exotic superconductivity^{24}. Finally, we explicitly demonstrate the SO(3) character of the vortices by enacting a change of basis, which appears experimentally as a spatial separation of phase singularities in the three spinor components. Our work directly addresses the challenges of controlled creation and simple parameter tuning of a fully 3D, singular SO(3) vortex, marking the path for a detailed study and direct imaging of the underlying topological phenomena.
Results
Theoretical background
The macroscopic wave function of a spin1 BEC can be written in terms of the atomic density n and the threecomponent spinor ζ as \(\Psi ({\mathbf{r}},t) = \sqrt {n({\mathbf{r}},t)} \zeta ({\mathbf{r}},t)\). In the FM phase, we have^{3}
which can be obtained by applying a 3D spin rotation U(α, β, γ) to the representative FM spinor (1, 0, 0)^{T}. Any FM spinor is thus fully specified by the three Euler angles α, β, and γ, corresponding to the group of rotations in three dimensions, SO(3). As a consequence, any FM state can be represented by the orientation of a vector triad defined by the condensate spin vector \(\langle {\hat{\mathbf{F}}}\rangle\) (\(F \equiv \langle {\hat{\mathbf{F}}}\rangle  = 1\)) and an orthogonal vector \({\hat{\mathbf{d}}}\) (Methods).
The topological stability of a singular SO(3) vortex is characterised by the way closed contours encircling the defect map into the order parameter space^{7}. If the order parameter space image of such a closed loop can be continuously contracted to a point, the defect is not topologically stable against transformations to the vortexfree state. The SO(3) parameter space may be represented geometrically as a solid sphere of radius π, where the direction of the radius vector of any point within the sphere gives an axis of rotation and its length gives the rotation angle (Fig. 2). However, π rotations about axes \({\hat{\mathbf{n}}}\) and \( {\hat{\mathbf{n}}}\) are equivalent, and thus diametrically opposite points on the surface must be identified. Therefore, only two topologically distinct classes of singular vortex lines exist: those that trace between identified, diametrically opposite points an even number of times, including zero; and those that trace between them an odd number of times. Mathematically, the vortex charges form the twoelement group \({\Bbb Z}_2\).
Since an even number of connections between identified points always corresponds to a loop contractible to a point, the vortices in the first (even) class can be continuously deformed into the defectfree state, and those in the second (odd) class can be continuously deformed to a singly quantised, singular vortex. The essence of Dirac’s belt trick is that a 4π winding, with a path in parameter space that goes about the sphere once, is equivalent to the defectfree state.
SO(3) vortex creation
Our primary result is a controlled creation method of a pair of singular SO(3) spinor vortices with nontrivial rotational topology from a nonsingular texture. In the initial nonsingular vortex—also known as a coreless vortex, baby skyrmion, or Anderson–Toulouse–Chechetkin/Mermin–Ho^{2} vortex in superfluid liquid helium—the circulation is not quantised and the spin forms a fountainlike profile that adjusts to the angular momentum of the superfluid. This characteristic fountain texture has been experimentally observed in BECs^{25,26,27}. If the nonsingular spin texture is not constrained, e.g., energetically, it can continuously deform to a vortexfree state. We find, however, that a very sharp bending of the vortex spin profile, corresponding to a strong but incomplete longitudinal magnetisation, induces an instability wherein the nonsingular spin texture decays by splitting into a pair of singly quantised vortices^{14}, as shown in Fig. 3a–d (see also Supplementary Note 1). Once separated, the resulting singly quantised vortices can no longer unwind on their own, thus circumventing the Dirac belt trick along the lines of Fig. 2. The decay paths of the nonsingular vortex therefore include not only its unwinding by local spin rotations or departure from the condensate at its boundary^{25}, but also its splitting into a pair of singly quantised SO(3) vortices that will, in turn, also ultimately leave the condensate. Numerically, a bending with magnetisation \(M \lesssim  0.3\) that is explicitly conserved is sufficient to guarantee the splitting, as shown in Figs. 3 and 4.
The splitting process of the nonsingular spin texture is fundamentally different from the previously observed decay of a multiply quantised singular vortex into multiple singly quantised vortices^{28,29,30,31}, in which magnetic trapping fields froze the atomic spin degree of freedom to produce a scalar BEC. In contrast, our experiment relies upon an alloptical trap that allows the atoms to retain their spinor nature. Even so, imprinting a multiply quantised singular vortex fully spinpolarises the condensate and spinor dynamics do not occur due to conservation of the maximised longitudinal magnetisation. The critical feature of our experiment is that the decay dynamics begin with an imprinted nonsingular spin texture. The incomplete magnetisation ensures active spin degrees of freedom, and a spinor description is required. The relevant algebra of the linevortex charges in our splitting process in SO(3) thus obeys the cyclic group \({\Bbb Z}_2\) with only the elements 0 and 1. Both evenly quantised and nonsingular vortices are represented by the trivial element and their splitting corresponds to the group operation 0 = 1 + 1, with no counterpart in a scalar BEC.
We use timevarying magnetic fields (Fig. 3e) to initiate the creation process experimentally with a condensate initially prepared in m = 1〉, where m〉 denotes the mth spinor component. Such techniques^{32,33} have been used to prepare, e.g., nonsingular^{25,27,34} and multiply quantised vortices^{29}, as well as monopoles^{35}, skyrmions^{36}, and knots^{37}.
Controlled creation of singular vortices in scalar BECs^{38,39} and continuous textures in spinor systems^{26} have also been achieved using phase imprinting methods. In our experiment the atoms experience an applied magnetic field described by
where b_{q} is the strength of the quadrupole contribution and B_{b}(t) is a timedependent bias field that shifts the location of the point at which the magnetic field vanishes (the field zero) to z_{0} = B_{b}/(2b_{q}) on the zaxis. We initially choose B_{b} such that the field zero is slightly above the condensate (see Methods) and the magnetic field is approximately uniform (Fig. 3e).
Reducing the bias field slowly induces adiabatic spin rotations as the magnetic field zero passes through the condensate from above, trailed by a 3D nodal line^{35} (Fig. 3e). At faster magnetic field ramp rates the otherwise identical experiment yields controllably incomplete adiabatic spin rotations, and results in a nonsingular vortex^{28,34} with additional populations in 0〉 and 1〉 (Fig. 3a, b and Supplementary Note 1). The atoms are released from the trap after an evolution time T_{evolve}, measured from the completion of the field ramp. Following a period of ballistic expansion they are imaged, whereupon we observe a pair of singly quantised SO(3) vortices in −1〉 with filled cores containing atoms in 0〉, as shown in Figs. 3c, d and 4. These results agree with a numerical simulation of the locally relaxed state (Supplementary Note 1). One of these singular spinor vortices typically departs the condensate before the other, thus lowering the condensate energy^{12,14} and leaving behind a single SO(3) vortex (Fig. 5). The main dissipative sources, as in scalar BECs^{6,40}, are a nonvanishing thermal cloud and potential collisions with hightemperature atoms.
Vortex core filling and interface
For comparison, we also produce vortices with empty cores by reducing the ramp rate such that the spins rotate nearly adiabatically, leaving the system with unobservable populations in 0〉 and 1〉. The size of the filled vortex core is typically much larger than that of an empty core, as shown in Fig. 6. We have numerically verified that for our experimental parameters the superfluid vortex core expands at a rate similar to that of the whole condensate after the release from the trap. In the experiment, the size of the filled vortex core is a further manifestation of the topology of the spinor where the spinor interactions break the \(\langle {\hat{\mathbf{F}}}\rangle  = 1\) spin condition of the FM phase. In the ground state, the size of a filled vortex core is determined by a spin healing length^{12,15} arising only from the spin–spin interactions, which is much larger than the density healing length that limits the size of an empty core. Thus, as the condensate evolves, dissipation causes the filled vortex cores to inflate as 0〉 atoms accumulate there. We observe no corresponding growth of empty vortex cores, as also shown in Fig. 6.
Whereas the SO(3) order parameter of the FM phase may be represented by the orientation of an orthonormal vector triad, the P order parameter is characterised by an unoriented nematic axis \({\hat{\mathbf{d}}}\) together with the condensate phase (Supplementary Note 2). The filling of the vortex core thus results in an interface between regions where the superfluid order parameter breaks different symmetries. In our system the interface appears in the internal structure of the defect itself and is observed directly in the experiment as a smooth transition between the FM vortex state in the surrounding superfluid and the P phase at the vortex core (Fig. 5). A numerical simulation of this transition allows us to portray the condensate spinor graphically in terms of a sphericalharmonic expansion, Z (see Fig. 1). The deformation of Z illustrates the continuous topological interface that connects the SO(3) symmetric order parameter of the FM phase to the nematic order parameter of the P phase. Note that in the pure FM phase, the triad order parameter corresponds exactly to the orientation and argument of Z.
Analytically, the spinor describing the vortex and its superfluid core can be constructed as an interpolating filledcore vortex solution as in ref. ^{14},
where \(D_ \pm = \sqrt {1 \pm F}\) represents the interpolation between the FM and P phases for F varying from 1 to 0, respectively. The azimuthal angle around the vortex line is represented by ϕ, and β is the polar angle. The spin vector is \(\langle {\hat{\mathbf{F}}}\rangle = F({\rm{sin}}\beta \hat {\boldsymbol{\rho }} + {\rm{cos}}\beta {\hat{\mathbf{z}}})\), and the unit vector orthogonal to it is \({\hat{\mathbf{d}}} =  {\rm{cos}}\beta \hat {\boldsymbol{\rho }} + {\rm{sin}}\beta {\hat{\mathbf{z}}}\), where \(\hat {\boldsymbol{\rho }}\) is the radial unit vector relative to the vortex line. For F = 1, Eq. (3) reduces to the singular FM vortex, and for F = 0, the spinor represents the noncirculating P phase that occupies the vortex core.
Spinor analysis
Next, we explicitly demonstrate the SO(3) nature of the vortex. The representation of the vortex wave function as a threecomponent spinor depends on the choice of the spinor basis, and the order parameter symmetry dictates how the representation transforms under a change of basis. Experimentally it is more convenient to change the orientation of the spin with respect to a fixed quantisation axis by applying a radiofrequency (RF) π/2 pulse, which rotates the spin according to the unitary transformation U(0, π/2, γ_{0}) where the arbitrary angle γ_{0} does not affect the outcome. The resulting density profiles are notably more complicated, as shown in Fig. 7. To understand these results theoretically, we assume cylindrical symmetry and neglect any small population in 1〉, leading to a qualitative model for the vortex
where \(g(\rho ) = \rho ^2/(\rho ^2 + r_0^2)\) approximates the vortexcore profile with size parameterised by r_{0}. The FM part of the spinor (4) in the original basis transforms as \(e^{i\phi }(0,0,1)^{\mathrm{T}} \to e^{i\phi }(1/2,  1/\sqrt 2 ,1/2)^{\mathrm{T}}\), distributing the atoms across all three components. The P part transforms as \((0,1,0)^{\mathrm{T}} \to (  1/\sqrt 2 ,0,1/\sqrt 2 )^{\mathrm{T}}\), splitting the atoms evenly between the ±1〉 components. Thus, after the pulse, the original atomic density distribution of the FM phase is reproduced in the 0〉 component as it only contains atoms that originated in the −1〉 component. On the other hand, the other two components exhibit phase singularities that have shifted to different locations, leading to a splitcore solution that appears to have broken the axial symmetry of the original state. This translation of the vortices after the basis transformation is a manifestation of the SO(3) symmetry of the order parameter, and indicates the presence of a line singularity about which the spin vector rotates (disgyration). After the π/2 rotation, one can still identify the locations of the vortices by the density minima of the atoms in the 0〉 component.
The matter wave in ±1〉 may also be interpreted as an interference between the overlapping spinor components before the spintip pulse. In all cases, the experimental density profiles of Fig. 7 agree well with the theoretical prediction obtained by applying a π/2 spin rotation to Eq. (4).
Discussion
Our results advance the experimental and theoretical investigations of defects containing topological interfaces. Similar techniques can be used to generate halfquantum vortices, as well as vortices with coherent interfaces involving the many diverse magnetic phases observed in spin2 spinor condensates^{41,42,43,44}. The filled vortex cores themselves may be used as tracers to examine the longitudinal dynamics of the vortex lines^{45}, which are otherwise difficult to discern. A further exciting extension would be to study the corresponding system in rotation where the nucleation and stability of vortices should dramatically depend on the precise value of the conserved magnetisation^{14}—determining whether nonsingular or singular vortices will prevail.
Methods
Experiment
The experimental techniques resemble those described in ref. ^{35}, beginning with an optically trapped ^{87}Rb condensate prepared in the FM phase (1, 0, 0)^{T} = 1〉. The optical trap frequencies are ω_{r} ≃ 2π × 130 Hz and ω_{z} ≃ 2π × 170 Hz in the radial and axial directions, respectively, and with an initial atom number N of typically 2 × 10^{5}. The axial Thomas–Fermi radius of the condensate is 5 μm and the corresponding radial extent is 7 μm. The bias magnetic field B_{b} is controlled by a single Helmholtz coil pair, and the quadrupole magnetic field strength b_{q} by a second coaxial antiHelmholtz pair. Two other pairs of coils for the x and y directions null those field components such that the field zero passes through the centre of the condensate.
The magnetic field zero is initially placed approximately 35 μm above the condensate with an initial gradient strength b_{q} = 4.3(4) G cm^{−1} and initial bias field B_{b} ≈ 30 mG. The bias field is then reduced to ~−50 mG at the rate dB_{b}/dt, and then to −0.38 G over the following 10 ms. The atoms are then held in the trap for a time T_{evolve}. An optional 8 μs, 0.266 MHz RF π/2 spintip pulse is applied immediately afterwards. At the conclusion of the experiment the quadrupole field and the optical trap are extinguished. A brief exposure to a magnetic field gradient of 70 G cm^{−1} during the 23 ms expansion separates the spinor components horizontally, after which they are imaged absorptively along the y and zaxes in a 0.1 G field aligned with the zaxis. Atom loss during the experiment, both during the ramp and during the subsequent evolution time, reduces the total number of atoms to approximately 2 × 10^{5} at the time of imaging.
Reducing the bias field at the rate −0.25 G s^{−1} results in a doublyquantised vortex in −1〉 and essentially no atoms in the other spinor components. The experiments with filled cores were conducted at higher ramp rates, between −4 and −6 G s^{−1}. Ramp rates exceeding −10 G s^{−1} result in larger nonsingular vortices that occupy all three spinor components. These are not observed to evolve into singular SO(3) vortices.
Numerical model
We use experimental parameters for the GrossPitaevskii Hamiltonian density of the spin1 BEC
where \(h_0 = \frac{{\hbar}^{2}}{2M_a}\left {\nabla{\Psi}} \right^2 + V({\mathbf{r}})n\) includes the harmonic trapping potential V(r). Here ħ is the reduced Planck constant and M_{a} is the atomic mass. The spin is defined as the expectation value \(\langle {\hat{\mathbf{F}}}\rangle = \mathop {\sum}\nolimits_{\alpha \beta } \zeta _\alpha ^\dagger {\hat{\mathbf{F}}}_{\alpha \beta }\zeta _\beta\), where \({\hat{\mathbf{F}}}\) is a vector of dimensionless spin1 Pauli matrices. The condensate spin vector corresponding to Eq. (1) is given by \(\langle {\hat{\mathbf{F}}}\rangle = {\rm{cos}}\alpha {\rm{sin}}\beta {\hat{\mathbf{x}}} + {\rm{sin}}\alpha {\rm{cos}}\beta {\hat{\mathbf{y}}} + {\rm{cos}}\beta {\hat{\mathbf{z}}}\). The FM order parameter can be defined by the orientation of two orthogonal vectors \(\langle {\hat{\mathbf{F}}}\rangle\) and \({\hat{\mathbf{d}}} = (  {\rm{sin}}\alpha {\rm{cos}}\gamma  {\rm{cos}}\alpha {\rm{sin}}\gamma {\rm{cos}}\beta ){\hat{\mathbf{x}}} + ({\rm{cos}}\alpha {\rm{cos}}\gamma  {\rm{sin}}\alpha {\rm{sin}}\gamma {\rm{cos}}\beta ){\hat{\mathbf{y}}} + {\rm{sin}}\gamma {\rm{sin}}\beta {\hat{\mathbf{z}}}\). The last two terms of Eq. (5) describe the linear and quadratic Zeeman shift of strengths p and q, respectively. The two interaction terms of strengths c_{0} and c_{2} arise from swave scattering of the atoms.
In swave scattering the only spinflip processes are \(2\left {m = 0} \right\rangle \rightleftharpoons \left {m = + 1} \right\rangle + \left {m =  1} \right\rangle\). The longitudinal magnetisation
where F_{z} is the z component of the condensate spin, is therefore approximately conserved on time scales for which swave scattering dominates. This condition is broken when the Gross–Pitaevskii equations are made dissipative, e.g., by imaginarytime evolution. We employ an algorithm to strictly restore the conservation of magnetisation^{14} throughout energy relaxation in pure imaginary time evolution and in evolution dynamics following imprinting, in which case we set time to include a small imaginary component t → (1 − iη)t, where η ~ 10^{−2}. All numerical simulations are carried out using a splitstep algorithm on a minimum of 128 × 128 × 128point grid.
Data availability
All relevant data sets generated during and/or analysed during the current study are available from the corresponding author upon request. The source data underlying Figs. 1 and 3–7 are provided as a Source Data file in the Zenodo repository (https://doi.org/10.5281/zenodo.3404017)^{46}.
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Acknowledgements
We acknowledge funding by the National Science Foundation (grant nos. PHY–1519174 and PHY–1806318), by the Emil Aaltonen Foundation, by the Kaupallisten ja teknillisten tieteiden tukisäätiö (KAUTE) foundation through its Researchers Abroad programme, by the Academy of Finland through the Centre of Excellence Programme (grant no. 312300) and ERC Consolidator Grant QUESS (no. 681311), and by the EPSRC.
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L.S.W., A.B., T.O. and D.S.H. developed and conducted the experiments and analysed the data. J.R. proposed the experiment. Numerical simulations were carried out by M.O.B., and the theoretical analysis was developed by M.O.B. and J.R. The manuscript was written by M.O.B., J.R. and D.S.H. All authors discussed the results and commented on the manuscript.
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Weiss, L.S., Borgh, M.O., Blinova, A. et al. Controlled creation of a singular spinor vortex by circumventing the Dirac belt trick. Nat Commun 10, 4772 (2019). https://doi.org/10.1038/s41467019127871
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