Current Address: ]Department of Electrical and Computer Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA

Current Address: ]Department of Physics and James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA

Vortices in superfluids with internal degrees of freedom, such as those existing within spinor Bose–Einstein condensates (BECs) Kawaguchi and Ueda (2012); Stamper-Kurn and Ueda (2013) and superfluid liquid He-3 Volovik (2003); Vollhardt and Wölfle (1990), exhibit a much richer phenomenology than do simple line vortices in scalar superfluids. Notable examples abound, including vortices with fractional charges Leonhardt and Volovik (2000); Ruostekoski and Anglin (2003); Ji et al. (2008); Lovegrove et al. (2012); Seo et al. (2015); Autti et al. (2016); Kang et al. (2019), vortices with like charges that sum to zero Weiss et al. (2019), vortices with charges that do not commute Kobayashi et al. (2009); Borgh and Ruostekoski (2016); Mawson et al. (2015); Semenoff and Zhou (2007); Barnett et al. (2007); Borgh et al. (2017); Mawson et al. (2019), and nonsingular textures with angular momentum Mizushima et al. (2002); Martikainen et al. (2002); Leanhardt et al. (2003); Leslie et al. (2009); Choi et al. (2012); Lovegrove et al. (2014). These features, inter alia, hint at their highly counter-intuitive dynamics.

The symmetry properties of the superfluid order parameter determine its magnetic phases and topologically permissible vortex excitations Kawaguchi and Ueda (2012); sup . The ground state of a spin-1 system, for example, exhibits two phases: a polar phase, which minimizes the total spin and is characterized by a nematic axis $\mathbf{\hat{d}}$ and condensate phase $\tau$; and a ferromagnetic (FM) phase, which maximizes the total spin and is characterized by a vector triad. In turn, the ground-state phase of an atomic BEC at zero magnetic field is determined by the nature of the interatomic interactions, which are themselves polar (e.g., in ²³Na) or FM (e.g., in ⁸⁷Rb) Kawaguchi and Ueda (2012). Thus do interactions at the atomic scale influence both the type and destiny of vortices within the condensate.

In the polar phase, a singly-quantized vortex (SQV) with $2\pi$ phase winding is unstable against splitting into a pair of half-quantum vortices (HQVs), each with $\pi$ phase winding. This unusual possibility was proposed and analyzed in Ref. Lovegrove et al. (2012) and subsequently observed experimentally within a ²³Na BEC in an effectively two-dimensional trapping geometry Seo et al. (2015, 2016). The core of a vortex in superfluid ³He-$B$ has similarly been predicted Thuneberg (1986); Salomaa and Volovik (1986) and observed Kondo et al. (1991) to consist of two HQVs; and, more recently, HQVs have been observed in the ³He polar phase Autti et al. (2016).

In this Letter we describe the controlled creation and subsequent time-evolution of a pair of three-dimensional (3D) singular SQVs in the polar phase of a spin-1 ⁸⁷Rb BEC with FM interatomic interactions. In contrast to techniques that randomly nucleate vortices throughout the superfluid by, e.g., stirring Rosenbusch et al. (2002); Neely et al. (2010); Seo et al. (2015) or rapid cooling through the superfluid transition Tilley and Tilley (1990); Weiler et al. (2008); Freilich et al. (2010), our experiment makes use of a deliberately applied strong bending of a nonsingular spin texture to generate a single pair of $\mathrm{SO}(3)$ vortices with polar cores at a specific location within the BEC Weiss et al. (2019). A sudden exchange of the polar and FM phases results in the desired pair of polar SQVs, where the topological interface Borgh and Ruostekoski (2012); Lovegrove et al. (2016) between the two magnetic phases within each vortex core is imaged directly. In a final step, a radio-frequency $\pi/2$ spinor rotation causes each SQV to evolve towards a pair of HQVs. We numerically model these experimental conditions and show how the FM interactions influence and complicate the decay process as compared with polar interactions.

The theoretical analysis uses the mean-field model for a spin-1 BEC, with Hamiltonian density Kawaguchi and Ueda (2012); Stamper-Kurn and Ueda (2013)

for the spinor wavefunction

expressed in a basis quantized along the $z$ axis. Here, $\mathbf{\hat{F}}$ is the vector of spin-1 matrices, $h_{0}=\hbar^{2}/(2M)|\nabla\Psi|^{2}+(M\omega_{r}^{2}/2)(x^{2}+y^{2}+2z^{2})n$ for atomic mass $M$ and radial trap frequency $\omega_{r}$, and $n=\Psi^{\dagger}\Psi$ is the atomic density. The constants $c_{0}$ and $c_{2}$ parameterize the spin-independent and spin-dependent interaction strengths, respectively, and $p\propto|\mathbf{B}|$ and $q\propto|\mathbf{B}|^{2}$ give the linear and quadratic Zeeman energy shifts due to an applied magnetic field $\mathbf{B}$.

The ground state of the system is determined by the sign of the interaction strength $c_{2}$ and, at fixed magnetization, the quadratic Zeeman term $q$ sup . In our experiment $q>|c_{2}|n$, specifying an easy-axis polar (EAP) ground-state phase Kawaguchi and Ueda (2012). We shall see that this introduces significant dynamics when $\mathbf{\hat{d}}$ is not aligned with the magnetic field.

The experiment begins with an FM ⁸⁷Rb condensate of $N\sim 2.0\times 10^{5}$ atoms in an optical trap with frequencies $(\omega_{r},\omega_{z})=2\pi(130,170)~{}\mathrm{s}^{-1}$. The atoms are exposed to a magnetic field described by

where $B_{z}$ is the strength of an applied bias field along the $z$ axis and $b_{q}$ is the strength of a 3D quadrupole field produced by a pair of anti-Helmholtz coils. The condensate spin is initially aligned along the $z$ axis, represented by the spinor $(1,0,0)^{\mathrm{T}}$ in the space-fixed basis we adopt for the remaining discussion.

We use a phase imprinting process to introduce the polar SQVs, initially following the vortex creation technique introduced in Ref. Weiss et al. (2019). A nonsingular vortex is first created by linearly ramping the magnetic bias field $B_{z}$ from $0.03$ G to $-0.05$ G at $-5$ G/s, with $b_{q}(0)=4.3(4)$ G/cm. The atomic spins incompletely follow the nonadiabatic reorientation of the magnetic field Nakahara et al. (2000); Pietilä and Möttönen (2009) as its zero passes through the condensate, resulting in the desired spin texture. Immediately afterwards we ramp the field to its minimum value $-0.38$ G in 10 ms, eliminate the magnetic quadrupole contribution $b_{q}\rightarrow 0$, and adiabatically reorient the field to 1 G along the $+z$ axis. In the subsequent 100 ms, the tight bending of the magnetization causes the nonsingular vortex to decay into two singular $\mathrm{SO}(3)$ vortices in the FM phase, each described in this basis by $(e^{i\varphi},0,0)^{\mathrm{T}}$, taking $\varphi$ to be the azimuthal angle around each vortex line.

In scalar superfluids a singular defect implies that the superfluid density at the singularity vanishes, but in spinor BECs it is energetically favorable to accommodate the singularity by filling the vortex core with atoms in a different magnetic phase when the spin-dependent interaction is weaker than the spin-independent one Ruostekoski and Anglin (2003); Lovegrove et al. (2012); Kobayashi et al. (2012). The vortex core regions in our experiment contain atoms in the nonrotating polar phase, described by $(0,1,0)^{\mathrm{T}}$, as shown in Fig. 1(b). These exhibit a coherent, stable topological interface between the two distinct magnetic phases Borgh and Ruostekoski (2012); Lovegrove et al. (2016), where the magnetic phase changes continuously within the vortex core. Analogous topological interfaces are universal across many areas of physics, ranging from superfluid liquid ³He Finne et al. (2006); Bradley et al. (2008) to early-universe cosmology and superstring theory Kibble (1976); Sarangi and Tye (2002), as well as to exotic superconductivity Bert et al. (2011).

The next step in the SQV creation process is the rapid exchange of the polar and FM phases. For the texture described above this amounts to swapping the $m=+1$ and $m=0$ spinor components with a sequence of three microwave pulses, as shown in Fig. 1(a). Afterwards, the topological interface between magnetic phases in the vortex core is reversed: along each singularity the atoms attain the pure FM phase, $(1,0,0)^{\mathrm{T}}$, while the surrounding circulating bulk superfluid is in the polar phase, $(0,e^{i\varphi},0)^{\mathrm{T}}$. These features are clearly seen in Fig. 1(c). Creation of this novel vortex state using the magnetic phase exchange technique is one of the principal results of the present study.

We model the two polar SQVs and their FM cores numerically in 3D [Fig. 2(a)]. Each vortex in the figure represents a polar SQV with a filled core. Analytically, the spinor representing both the vortex and its FM core can be constructed as Lovegrove et al. (2016)

where $D_{\pm}=(1\pm|\langle\mathbf{\hat{F}}\rangle|)^{1/2}$ parameterizes the interpolation between the polar and FM phases as $|\langle\mathbf{\hat{F}}\rangle|$ varies from $0$ in the bulk to $1$ on the vortex line. Here $\varphi$ again denotes the azimuthal angle around the vortex line, whereas $\beta$ is the polar angle that determines the order-parameter orientation, varying from $\beta=\pi/2$ away from the vortex line to $\beta=0$ on the line singularity itself. The initial state is modeled numerically and shown in Fig. 2(a,b).

Although the spontaneous breaking of the defect core symmetry in the polar phase and the emergence of HQVs depend nontrivially on the relative interaction strength $c_{2}/c_{0}$ Ruostekoski and Anglin (2003); Shinn and Fischer (2018); Underwood et al. (2020), this dependence can easily be obscured by the density gradients in a harmonically trapped BEC. Of additional significance is the effect of the applied magnetic field, which can restore the vortex core isotropy at sufficiently high $p$ and suppress the decay into HQVs at sufficiently high $q$ Borgh et al. (2016); Underwood et al. (2020). We find empirically that an applied bias magnetic field of 1 G is sufficient to inhibit evolution of the experimental SQV state depicted in Fig. 1(c) towards HQVs.

To induce condensate dynamics, we therefore apply a $\pi/2$-pulse within the $F=1$ manifold to rotate both the nematic director (in the polar phase) and the condensate spin (in the FM phase) into the $xy$ plane. The spinor rotation can be understood by describing a single SQV as a vortex line in the $m=0$ component with core filled by atoms in the $m=+1$ component. Under a $\pi/2$ spinor rotation about the $y$ axis, the spinor transforms as

where $g(\rho)=\rho^{2}/(\rho^{2}+r_{0}^{2})$ approximates the vortex-core profile with size parameterized by $r_{0}$. After the rotation, density maxima in the $m=0$ component appear at the locations of the vortex cores, each bracketed by symmetrically displaced density minima in the $m=\pm 1$ spinor components. These spatially offset phase singularities result from the sum of FM and polar terms of the initial spinor that transform differently within the vortex core Weiss et al. (2019); sup . We experimentally observe these principal features immediately after the $\pi/2$ pulse [Fig. 3(a)].

As the system evolves, the spatial separation between the phase singularities associated with both SQVs increase, and each phase singularity grows in size as it fills with fluid in the $m=+1$ (or, for the other of the pair, $m=-1$) spinor component (Fig. 3). These effects can be seen in the emergence of sharply defined proximate bright and dark regions in the total longitudinal magnetization density $M(\mathbf{r})\approx n_{+1}-n_{-1}$ sup , shown in the rightmost column of Fig. 3, as well as directly from the locations of the density minima within the $m=\pm 1$ spinor components. The density of the $m=0$ spinor component also becomes more diffuse as the core region of each SQV grows.

We numerically simulate the dynamics using the coupled Gross–Pitaevskii equations derived from Eq. (1), also employing an algorithm to restore the conservation of longitudinal magnetization Lovegrove et al. (2014) in the presence of a small phenomenological dissipation. The resulting evolved state is shown in Fig. 2(e–f) where the pair of phase singularities associated with each SQV has formed a split pair of HQVs. In the simulation, the fully separated FM cores and the symmetry representation of the wavefunction shown in Fig. 2(e–f) permit straightforward identification of the HQVs. Once separated, the characteristic size of the filled regions is theoretically established by the spin healing length Lovegrove et al. (2012); Ruostekoski and Anglin (2003), which is much larger than the density healing length.

Connecting the experimentally obtained spinor component densities to the corresponding vortex states in the simulation requires some care, especially with respect to the presence of the $m=0$ spinor component. For the given magnetic field direction and the idealized case of a condensate with polar interactions in the easy-plane polar regime sup , an empty-core SQV splits into two spatially offset phase singularities in the $m=\pm 1$ spinor components and the $m=0$ component remains absent. These phase singularities are unambiguously HQVs, each surrounded by polar fluid where $\mathbf{\hat{d}}$ remains in the plane. For FM interactions in the EAP regime, however, the presence of the phase singularities in regions where the $m=0$ component is nonzero can indicate either the existence of nonzero transverse spin within the unsplit SQV core [Fig. 2(c–d)], or the rotation of $\mathbf{\hat{d}}$ out of the $xy$ plane in a fully split pair of HQVs [Fig. 2(e–f)]. Only the disappearance of the $m=0$ spinor component in the experimental images, implying fully longitudinal spin domains with a director that remains in the $xy$ plane, conclusively announces the presence of two HQV. This is approximately the situation in Fig. 3(e–f).

There are several effects that conspire to complicate the experimental interpretation of our results. First, the offset phase singularities of the two initial singular vortices may closely approach one another, making their disambiguation problematic. The approximate SQV locations may often be located by identifying density maxima in the $m=0$ spinor component, as shown in Fig. 3(a–d). The presence of a small and consistent but uncontrolled magnetic field gradient globally shifts the $m=\pm 1$ spinor components with respect to one another, creating less sharply defined regions of opposite magnetization on a size scale comparable to that of the condensate. This effect is most pronounced in Fig. 3(c,d), where the $m=+1$ ($m=-1$) spinor component is shifted to the right (left). Nevertheless, the vortices may often be located by identifying the density minima associated with the offset phase singularities in the $m=\pm 1$ spinor components, as suggested by the dashed circles in Fig. 3(e) and (f). The singularities can still be difficult to discern if they are near the edge of the condensate, if they are tilted with respect to the imaging axis, or if they exhibit longitudinal (Kelvin wave) excitations Bretin et al. (2003). One of the expected singularities in Fig. 3(e) likely cannot be cleanly identified as a result of one or more of these 3D effects.

We have implemented a controllable technique of rapid magnetic phase exchange to facilitate the controlled creation of a pair of singular SQVs with nonrotating FM cores in the polar magnetic phase of a spin-1 superfluid with FM interatomic interactions. Our experimental and theoretical analysis of the decay process in three dimensions suggests the emergence of HQVs. Similar techniques may be used to generate pairs of filled-core vortices in the magnetic phases of spin-2 condensates, where vortex collisions are predicted to possess a non-Abelian character Kobayashi et al. (2009); Borgh and Ruostekoski (2016); Semenoff and Zhou (2007); Mawson et al. (2019) and for which the topological interfaces may lead to exotic phenomena such as vortices with triangular cores Borgh and Ruostekoski (2016).