Abstract
Quantized vortices appear in physical systems from superfluids and superconductors to liquid crystals and high energy physics. Unlike their scalar cousins, superfluids with complex internal structure can exhibit rich dynamics of decay and even fractional vorticity. Here, we experimentally and theoretically explore the creation and time evolution of vortex lines in the polar magnetic phase of a trapped spin1 ^{87}Rb Bose–Einstein condensate. A process of phaseimprinting a nonsingular vortex, its decay into a pair of singular spinor vortices, and a rapid exchange of magnetic phases creates a pair of threedimensional, singular singlyquantized vortex lines with core regions that are filled with atoms in the ferromagnetic phase. Atomic interactions guide the subsequent vortex dynamics, leading to core structures that suggest the decay of the singlyquantized vortices into halfquantum vortices.
Introduction
Vortices in superfluids with internal degrees of freedom, such as those existing within spinor Bose–Einstein condensates (BECs)^{1,2} and superfluid liquid He3^{3,4}, exhibit a much richer phenomenology than do simple line vortices in scalar superfluids. Notable examples abound, including vortices with fractional charges^{5,6,7,8,9,10,11}, vortices with like charges that sum to zero^{12}, vortices with charges that do not commute^{13,14,15,16,17,18,19}, and nonsingular textures with angular momentum^{20,21,22,23,24,25}. These features, among others, hint at their highly counterintuitive dynamics.
The symmetry properties of the superfluid order parameter determine its magnetic phases and topologically permissible vortex excitations^{1}. The ground state of a spin1 system, for example, exhibits two phases: a polar phase, which minimizes the total spin and is characterized by a nematic axis \(\hat{{{{\bf{d}}}}}\) and condensate phase τ; and a ferromagnetic phase, which maximizes the total spin and is characterized by a vector triad. In turn, the groundstate phase of an atomic BEC in zero magnetic field is determined by the nature of the interatomic interactions, which are themselves polar (e.g., in ^{23}Na) or ferromagnetic (e.g., in ^{87}Rb)^{1}. Interactions at the atomic scale thus influence both the type and destiny of vortices within the condensate.
In the polar phase, a singlyquantized vortex (SQV) with 2π phase winding is unstable against splitting into a pair of halfquantum vortices (HQVs), each with π phase winding. This unusual possibility was proposed and analyzed in ref. ^{8} and subsequently observed experimentally within a ^{23}Na BEC in an effectively twodimensional trapping geometry^{9,26}. The core of a vortex in superfluid ^{3}HeB has similarly been predicted^{27,28} and observed^{29} to consist of two HQVs; and, more recently, HQVs have been observed in the ^{3}He polar phase^{10}.
Here, we describe the controlled creation and subsequent time evolution of a pair of threedimensional (3D) singular SQVs in the polar phase of a spin1 ^{87}Rb BEC with ferromagnetic interatomic interactions. In contrast to techniques that randomly nucleate vortices throughout the superfluid by, e.g., stirring^{9,30,31} or rapid cooling through the superfluid transition^{32,33,34}, our experiment makes use of a deliberately applied strong bending of a nonsingular spin texture to generate a pair of singular, singlyquantized ferromagneticphase [SO(3)] vortices with polarphase filled cores at a specific location within the BEC^{12}. We then swiftly exchange the polar and ferromagnetic phases using a trio of microwave transitions that appropriately reapportion the spinor components of the system. The resulting pair of polar SQVs possesses a topological interface^{35,36} between the two magnetic phases within each vortex core that we image directly. In a final step, a radiofrequency π/2 spinor rotation causes each SQV to evolve toward a pair of HQVs. Our creation technique is also distinct from phaseimprinting methods via optical or microwave transitions^{23,37,38} or subjecting condensates to artificial gauge fields^{39}. We numerically model the experimental conditions of vortex preparation and show how the ferromagnetic interactions influence and complicate the decay process as compared with polar interactions.
Results and discussion
Creation of singular vortices
The BEC wavefunction Ψ(r, t) is expressed in terms of the atomic spinor ζ(r, t) and atom density n(r, t) in a basis quantized along the zaxis as
The ground state of the system is determined by the sign of the spindependent interaction strength c_{2} (“Methods”) and the quadratic Zeeman term q, as the linear Zeeman term p does not participate in a system with fixed longitudinal magnetization. In our experiments we have q > ∣c_{2}∣n, specifying an easyaxis polar groundstate phase for ^{87}Rb^{1}. In comparison to experiments involving ^{23}Na^{9,26}, for which q < ∣c_{2}∣n and the groundstate phase is easyplane polar^{1}, we shall see that this introduces significantly different dynamics when \(\hat{{{{\bf{d}}}}}\) (“Methods”) is not aligned with the magnetic field.
The experiment begins with an optically trapped ferromagnetic ^{87}Rb condensate of N ~ 2.0 × 10^{5} atoms (“Methods”). The atoms are exposed to a magnetic field described, in Cartesian coordinates (x, y, z) with respect to the center of the condensate, by
where B_{z} is the strength of an applied bias field along the zaxis produced by a Helmholtz coil pair, and b_{q} is the strength of a 3D quadrupole field produced by a coaxial pair of antiHelmholtz coils. Two other pairs of coils aligned with the x and y directions null those bias field components. The condensate spin is initially aligned along the zaxis, represented by the ferromagnetic spinor (1, 0, 0)^{T} in the spacefixed basis we adopt for the remaining discussion. At the conclusion of the experiments, described below, the quadrupole field and the optical trap are extinguished, and the spinor components are separated using a Stern–Gerlach separation technique (“Methods”). The atomic density distributions are then simultaneously imaged along the y and zaxes in a magnetic field aligned with the zaxis.
We use a phaseimprinting process to introduce the polar SQVs, following the vortex creation technique introduced in ref. ^{12}. A nonsingular vortex is first created by linearly ramping the magnetic bias field B_{z} to bring the location at which B vanishes, at z = B_{z}/(2b_{q}), entirely through the condensate along the zaxis (“Methods”). The atomic spins incompletely follow this field reorientation^{40,41}, permitting us to tailor the bending of the magnetization density by selecting the rate at which the zero of the field passes through the condensate. Instead of creating a nonsingular vortex representing a continuous, smooth spin texture, our specific choice triggers an instability that causes the nonsingular vortex to decay into two singular SO(3) vortices in the ferromagnetic phase^{12}, each described in this basis by \({({{{{\rm{e}}}}}^{{{{\rm{i}}}}\varphi },0,0)}^{{{{\rm{T}}}}}\) where φ is the azimuthal angle around the 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 spindependent interaction is weaker than the spinindependent one^{6,8,42}. The vortexcore regions in our experiment thus contain atoms in the nonrotating polar phase, described on this basis by (0, 1, 0)^{T}, as shown in Fig. 1. These exhibit a coherent, stable topological interface between the two distinct magnetic phases^{35,36,43}, where the magnetic phase changes continuously within the vortex core. Analogous topological interfaces are universal across many areas of physics, ranging from superfluid liquid ^{3}He^{44,45} to earlyuniverse cosmology and superstring theory^{46,47}, as well as to exotic superconductivity^{48}.
We use a magnetic phase exchange technique to rapidly exchange the polar and ferromagnetic phases. For the texture described above this amounts to swapping the m = +1 and m = 0 spinor components with a sequence of three resonant microwave πpulses through an intermediate state (“Methods”), as summarized in Fig. 1a. Afterward, the topological interface between magnetic phases in the vortex core is reversed: along each singularity, the atoms attain the pure ferromagnetic phase (1, 0, 0)^{T}, while the surrounding circulating bulk superfluid is in the polar phase \({(0,{{{{\rm{e}}}}}^{{{{\rm{i}}}}\varphi },0)}^{{{{\rm{T}}}}}\). These features are clearly seen in Fig. 1b–d. Creation of this vortex state using magnetic phase exchange is one of the principal results of the present study.
Vortex cores and dynamics
We model the two polar SQVs and their ferromagnetic cores numerically in 3D (Fig. 2a–f). Each vortex in the figure represents a polar SQV with a filled core. Analytically, the spinor representing both the vortex and its ferromagnetic core can be constructed as^{36}
where \({D}_{\pm }={(1\pm  \langle \hat{{{{\bf{F}}}}}\rangle  )}^{1/2}\) parameterizes the interpolation between the polar and ferromagnetic phases as \( \langle \hat{{{{\bf{F}}}}}\rangle \) varies from 0 in the bulk to 1 on the vortex line. Here φ again denotes the azimuthal angle around the vortex line, whereas β is the polar angle that determines the orderparameter orientation, varying from β = π/2 away from the vortex line to β = 0 on the line singularity itself. The initial state is modeled numerically and shown in Fig. 2a, 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}^{6}, 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 vortexcore isotropy at sufficiently high p and suppress the decay into HQVs at sufficiently high q^{49,50}. 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. 1c toward HQVs.
To induce condensate dynamics, we apply a π/2pulse within the F = 1 manifold to rotate both the nematic director (in the polar phase) and the condensate spin (in the ferromagnetic 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 π/2 spinor rotation about the y axis, the spinor transforms as
where \(g(\rho )={\rho }^{2}/({\rho }^{2}+{r}_{0}^{2})\) approximates the vortexcore profile with size parameterized by r_{0}. The transformation defined by Eq. (4) is illustrated in Figs. 2c, d and 3a–c. After the rotation, a density maximum in the m = 0 component appears at the location of the vortex core, bracketed by symmetrically displaced density minima in the m = ±1 spinor components. These spatially offset phase singularities result from the sum of ferromagnetic and polar terms of the initial spinor that transform differently within the vortex core^{12}. Note that the spinor wavefunction still contains single SQVs—the presence of the offset phase singularities alone does not imply splitting. In the experiment, we observe these principal features for each of the two created SQVs immediately after the π/2pulse (Fig. 4a).
As the system evolves, the spatial separation between the phase singularities associated with both SQVs increases, 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. 4a–f). These effects can be seen in the emergence of sharply defined proximate bright and dark regions in the total longitudinal magnetization density M(r) ≈ n_{+1} − n_{−1}, shown in the rightmost column of Fig. 4, as well as directly from the locations of the density minima within the m = ±1 spinor components. The density of the m = 0 spinor component also becomes more diffuse as the core region of each SQV grows. We stress that the observed vortices exist in a threedimensional condensate, as confirmed by the examples shown in Fig. 5a–d.
We numerically simulate the dynamics using the coupled Gross–Pitaevskii equations (see “Methods”), also employing an algorithm to restore the conservation of longitudinal magnetization^{25} in the presence of a small phenomenological dissipation. The resulting evolved state is shown in Fig. 2e, where the pair of phase singularities associated with each SQV has formed a split pair of HQVs. In the simulation, the fully separated ferromagnetic cores and the symmetry representation of the wavefunction shown in Fig. 2e, f permit straightforward identification of the HQVs. Once separated, the characteristic size of the filled regions is theoretically established by the spin healing length^{6,8}, which is much larger than the density healing length.
Experimental interpretation
Connecting the experimentally obtained spinorcomponent 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 pure energy relaxation in a condensate with polar interactions in the easyplane polar regime, an emptycore SQV is known to relax into a pair of HQVs^{8} (Fig. 6a, b). These are readily identified in the simulation by the orderparameter symmetry, also showing continuous deformation to the ferromagnetic vortex core. The vector \(\hat{{{{\bf{d}}}}}\) remains oriented in the xy plane everywhere, yielding a completely depopulated m = 0 spinor component and equal bulk densities in the m = ±1 components. The vortices appear as offset phase singularities in the populated components^{5}. In this case, the HQVs can conversely be inferred directly from the component images where the m = 0 spinorcomponent vanishes, as this corresponds directly to the zcomponent of \(\hat{{{{\bf{d}}}}}\).
For ferromagnetic interactions in the easyaxis polar 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. 6c, d), or the rotation of \(\hat{{{{\bf{d}}}}}\) out of the xy plane in a fully split pair of HQVs (Fig. 6e, f). Dissipation (included phenomenologically in the simulations; see “Methods”) causes the condensate to evolve toward the ferromagnetic phase; on the other hand, the Zeeman energy favors rotating \(\hat{{{{\bf{d}}}}}\) out of the xy plane. Both effects may lead to a nonzero density remaining in the m = 0 component. 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. 4e, f.
There are thus several effects that conspire to complicate the experimental interpretation of our results. First, the offset phase singularities of the two initial singular vortices might closely approach one another, making their disambiguation problematic. Second, the presence of a small and consistent but uncontrolled magnetic field gradient globally shifts the m = ±1 spinor components with respect to one another, creating less sharply defined regions of opposite longitudinal magnetization density on a size scale comparable to that of the condensate. This effect is pronounced in, e.g., Fig. 4c, d, where the m = +1 (m = −1) spinor component is shifted to the right (left). Nevertheless, the approximate vortex locations can often be determined by identifying either the density maxima in the m = 0 spinor component, as shown in Fig. 4a–d, or the density minima associated with the offset phase singularities in the m = ±1 spinor components, as suggested by the dashed circles in Fig. 4e, f. The singularities are still 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^{51}. One of the expected singularities in Fig. 4e is less cleanly identified as a result of one or more of these 3D effects.
Concluding remarks
We have implemented a controllable technique of rapid magnetic phase exchange to facilitate the controlled creation of a pair of singular SQVs with nonrotating ferromagnetic cores in the polar magnetic phase of a spin1 superfluid with ferromagnetic 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 filledcore vortices in the magnetic phases of spin2 condensates, where vortex collisions are predicted to possess a nonAbelian character^{13,14,16,19} and for which the topological interfaces may lead to exotic phenomena such as vortices with triangular cores^{14}.
Methods
Experimental details
The condensate begins in an optical trap with frequencies (ω_{r}, ω_{z}) = 2π(130, 170) s^{−1}. The axial Thomas–Fermi radius of the condensate is thus 5 μm and the corresponding radial extent is 7 μm. The condensate is exposed to a magnetic field described by Eq. (2), with B_{z}(0) = 0.03 G and b_{q}(0) = 4.3(4) G/cm, and is initially prepared in the ferromagnetic phase described by the spinor (1, 0, 0)^{T}. To create a nonsingular vortex we ramp B_{z} to ~ −0.1 G at −5 G/s. Immediately afterward we ramp the field to its minimum value −0.38 G in 10 ms, eliminate the magnetic quadrupole contribution b_{q} → 0, and adiabatically reorient the field to ~1 G along the +zaxis. Over the subsequent 100 ms, the tight bending of the magnetization density causes the nonsingular vortex to decay into two singular SO(3) vortices in the ferromagnetic phase^{12}.
Swapping the m = 0 and m = +1 spinor components achieves the exchange of the magnetic phases in this basis (Fig. 1). Our protocol first places the F = 1, m = 0 spinor component into the F = 2, m = 0 state using a 10 μs microwave πpulse at 6834.683197 MHz. The second microwave πpulse, 82 μs at 6835.391898 MHz, fully exchanges the populations in the F = 2, m = 0 state and the F = 1, m = 1 state. The final microwave πpulse is identical to the first, bringing the population in the F = 2, m = 0 state to the F = 1, m = 0 state. The optional radiofrequency π/2pulse is 15 μs long at 0.708774 MHz. These frequencies are based on an independent measurement of the magnetic field, which in this case is 1.009 G.
At the conclusion of the experiment, the atoms are released from the trap. A 4.2 ms exposure to a magnetic field gradient of ≈70 G cm^{−1} at the beginning of the subsequent 23 ms expansion separates the spinor components horizontally, after which they are imaged simultaneously along the y and zaxes.
Numerical model
The Gross–Pitaevskii Hamiltonian density of the spin1 BEC reads as^{1,2}
where we take \({h}_{0}=\frac{{\hslash }^{2}}{2{M}_{{{{\rm{a}}}}}} \nabla {{\Psi }}{ }^{2}+V({{{\bf{r}}}})n\) to include the harmonic trapping potential \(V({{{\bf{r}}}})=({M}_{{{{\rm{a}}}}}{\omega }_{r}^{2}/2)({x}^{2}+{y}^{2}+2{z}^{2})\). Here ℏ is the reduced Planck constant, M_{a} the atomic mass, and ω_{r} the radial trap frequency. The atomic density is n = Ψ^{†}Ψ, where the spinor wavefunction Ψ is defined in Eq. (1). The condensate spin is the expectation value \(\langle \hat{{{{\bf{F}}}}}\rangle ={\sum }_{\alpha \beta }{\zeta }_{\alpha }^{{\dagger} }{\hat{{{{\bf{F}}}}}}_{\alpha \beta }{\zeta }_{\beta }\), where \(\hat{{{{\bf{F}}}}}\) is a vector of dimensionless spin1 Pauli matrices. The constants c_{0} and c_{2} parameterize the spinindependent and spindependent interaction strengths, respectively, and p ∝ ∣B∣ and q ∝ ∣B∣^{2} give the linear and quadratic Zeeman energy shifts due to an applied magnetic field B. We employ experimental parameters in the coupled Gross–Pitaevskii equations derived from Eq. (5) to simulate the condensate dynamics. Dissipation is included phenomenologically by setting time to include a small imaginary component t → (1 − iη)t, where η ~ 10^{−2}.
The only spinflip processes allowed by angularmomentum conservation in swave scattering are \(2\leftm=0\right.\rangle \rightleftharpoons \leftm=+1\right.\rangle +\leftm=1\right.\rangle\). On time scales where swave scattering is dominant, this implies that the longitudinal magnetization
where F_{z} is the zcomponent of the condensate spin, is approximately conserved. Making the Gross–Pitaevskii equations dissipative breaks this conservation law. In our simulations, we therefore combine a splitstep method^{52} with an algorithm that explicitly conserves the longitudinal magnetization^{25}.
Magnetic phases
The two distinct groundstate phases in a spin1 Bose–Einstein condensate at zero field are determined by the sign of the spindependent atomic interaction parameter c_{2} in Eq. (1). For c_{2} < 0 (e.g., for ^{87}Rb) the system energy is minimized for \( \langle \hat{{{{\bf{F}}}}}\rangle  =1\), resulting in a ferromagnetic orderparameter space characterized by the group of 3D rotations, SO(3)^{53,54}. Contrariwise, for c_{2} > 0 (e.g., for ^{23}Na) the system energy is minimized for \( \langle \hat{{{{\bf{F}}}}}\rangle  =0\), resulting in a uniaxial nematic (polar) orderparameter space characterized by \([{S}^{2}\times {{{\rm{U(1)}}}}]/{{\mathbb{Z}}}_{2}\) with local U(1) phase τ and nematic axis \(\hat{{{{\bf{d}}}}}\)^{5,55,56}, defined implicitly by
The twoelement factor group \({{\mathbb{Z}}}_{2}\) appears due to the symmetry \(\hat{{{{\bf{d}}}}}{{{{\rm{e}}}}}^{{{{\rm{i}}}}\tau }=\hat{{{{\bf{d}}}}}{{{{\rm{e}}}}}^{{{{\rm{i}}}}(\tau +\pi )}\). It is this symmetry that permits vortices in the polar phase to carry halfinteger circulation when τ runs between 0 and π around the vortex singularity and \(\hat{{{{\bf{d}}}}}\) concurrently rotates by π.
At stronger fields with fixed longitudinal magnetization, the quadratic Zeeman term q becomes important for determining the ground state^{57,58,59,60}, and within the polar phase itself there arise two relevant phases: the easyaxis polar, where \(\hat{{{{\bf{d}}}}}\) is aligned with an applied magnetic field, and the easyplane polar, where \(\hat{{{{\bf{d}}}}}\) is perpendicular to the applied magnetic field. In our experiment q > ∣c_{2}∣n, specifying an easyaxis polar groundstate phase.
The orderparameter symmetry in Figs. 2 and 6 is given as the surface of ∣Z(θ, ϕ)∣^{2}, where \(Z(\theta ,\phi )=\mathop{\sum }\nolimits_{m = 1}^{+1}{Y}_{1,m}(\theta ,\phi ){\zeta }_{m}\) expands the spinor in terms of the spherical harmonics Y_{1,m}(θ, ϕ), parameterized by local spherical coordinates (θ, ϕ), with gauge color given by \({{{\rm{Arg}}}}(Z)\).
Code availability
The code is available upon reasonable request.
Change history
13 August 2021
A Correction to this paper has been published: https://doi.org/10.1038/s4200502100699w
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Acknowledgements
We gratefully acknowledge experimental assistance from and helpful conversations with T. Ollikainen. D.S.H. acknowledges financial support from the National Science Foundation (Grant No. PHY1806318.) and J.R. from the UK EPSRC (Grant Nos. EP/P026133/1, EP/S002952/1). The numerical computations were performed using the High Performance Computing Cluster supported by the Research and Specialist Computing Support service at the University of East Anglia.
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Y.X., L.S.W., A.A.B., and D.S.H. developed and conducted the experiments and analyzed the data. 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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Xiao, Y., Borgh, M.O., Weiss, L.S. et al. Controlled creation and decay of singlyquantized vortices in a polar magnetic phase. Commun Phys 4, 52 (2021). https://doi.org/10.1038/s4200502100554y
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DOI: https://doi.org/10.1038/s4200502100554y
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