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arXiv:1306.4700 · Physical Review Letters

Energetic Stability of Coreless Vortices in Spin-1 Bose-Einstein Condensates with Conserved Magnetization

Justin Lovegrove, Magnus O. Borgh, Janne Ruostekoski

Published Feb 19, 2014• 39 citations
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Abstract

We show that conservation of longitudinal magnetization in a spinor condensate provides a stabilizing mechanism for a coreless vortex phase-imprinted on a polar condensate. The stable vortex can form a composite topological defect with distinct small- and large-distance topology: the inner ferromagnetic coreless vortex continuously deforms toward an outer singular, singly quantized polar vortex. A similar mechanism can also stabilize a nonsingular nematic texture in the polar phase. A weak magnetization is shown to destabilize a coreless vortex in the ferromagnetic phase.

DOI
10.1103/physrevlett.112.075301

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We show that conservation of longitudinal magnetization in a spinor condensate provides a stabilizing mechanism for a coreless vortex phase-imprinted on a polar condensate. The stable vortex can form a composite topological defect with distinct small- and large-distance topology: the inner ferromagnetic coreless vortex continuously deforms toward an outer singular, singly quantized polar vortex. A similar mechanism can also stabilize a nonsingular nematic texture in the polar phase. A weak magnetization is shown to destabilize a coreless vortex in the ferromagnetic phase.
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.
We analyze the structure and stability of singular singly quantized vortices in a rotating spin-1 Bose-Einstein condensate. We show that the singular vortex can be energetically stable in both the ferromagnetic and polar phases despite the existence of a lower-energy nonsingular coreless vortex in the ferromagnetic phase. The spin-1 system exhibits an energetic hierarchy of length scales resulting from different interaction strengths, and we find that the vortex cores deform to a larger size determined by the characteristic length scale of the spin-dependent interaction. We show that in the ferromagnetic phase the resulting stable core structure, despite apparent complexity, can be identified as a single polar core with an axially symmetric density profile which is nonvanishing everywhere. In the polar phase, the energetically favored core deformation leads to a splitting of a singly quantized vortex into a pair of half-quantum vortices that preserves the topology of the vortex outside the extended core region, but breaks the axial symmetry of the core. The resulting half-quantum vortices exhibit nonvanishing ferromagnetic cores.
We show how a singly quantized vortex can exhibit energetically stable defect cores with different symmetries in an atomic spin-1 polar Bose-Einstein condensate, and how a stable topologically nontrivial Skyrmion texture of lower dimensionality can be confined inside the core. The core isotropy and the stability of the confined texture are sensitive to Zeeman level shifts. The observed structures have analogies, respectively, in pressure-dependent symmetries of superfluid liquid ^{3}He vortices and in the models of superconducting cosmic strings.
We theoretically investigate vortex-lattice phases of rotating spinor Bose-Einstein condensates (BEC's) with a ferromagnetic spin interaction by numerically solving the Gross-Pitaevskii equation. The spinor BEC under slow rotation can sustain a rich variety of exotic vortices due to the multicomponent order parameters, such as Mermin-Ho and Anderson-Toulouse coreless vortices (the two-dimensional Skyrmion and meron) and nonaxisymmetric vortices with shifting vortex cores. Here, we present the spin texture of various vortex-lattice states at higher rotation rates and in the presence of an external magnetic field. In addition, the vortex phase diagram is constructed in the plane of the total magnetization MMM and the external rotation frequency $\ensuremath{\Omega}$ by comparing the free energies of possible vortices. It is shown that the vortex phase diagrams in the $M\text{\ensuremath{-}}\ensuremath{\Omega}$ plane may be divided into two categories: (i) the coreless vortex lattice formed by the several types of Mermin-Ho vortices and (ii) the vortex lattice filling in the cores with the pure polar (antiferromagnetic) state. In particular, it is found that the type-(ii) state forms composite lattices of coreless and polar-core vortices. The difference between type (i) and type (ii) results from the existence of the singularity of the spin textures, which may be experimentally confirmed by the spin imaging within polarized light recently proposed by Carusotto and Mueller. We also discuss the stability of triangular and square lattice states for rapidly rotating condensates.
This thesis is concerned with the structure and stability of vortices in spin-1 atomic Bose-Einstein Condensates (BECs) in rotating, optical traps. We numerically study these vortex structures using a classical mean-field theory which allows atomic interactions to change the local expectation value of the atomic spin. Initially applying a model in which the atoms interact only via scattering which does not conserve an initial longitudinal magnetisation, we identify the energetically stable configurations of singular and nonsingular vortices via propagation in imaginary time in a rotating frame of reference. We find that the cores of singular vortices fill with atoms in the spinor BEC and show that this can be understood in terms of an energetic hierarchy of length scales. By refining the numerical model to explicitly conserve longitudinal magnetisation, we show that the conservation of a strong magnetisation can lead to a mixing of the two phases of the ground-state manifold (polar and ferromagnetic), which are characterised by the expectation value of the spin. This occurs as a result of the introduction of a new characteristic length scale determined by the longitudinal magnetisation. A surprising consequence is the stability of a ferromagnetic coreless vortex in the polar interaction regime, which otherwise is energetically unstable. We construct analytic spinor wavefunctions which parametrise the interpolation between the polar and ferromagnetic phases, exhibiting different vortex topologies in the respective phases. Finally by identifying stationary states of the system, we show how nonlocal dipoledipole interactions between atoms introduces an additional length scale determined by the strength of the dipolar interaction. The energetic hierarchy of length scales then determines whether the dipolar interactions have a significant effect on the stationary vortex structures. In particular we show how a BEC with polar interactions adopts the properties of a ferromagnetic condensate when the dipolar interactions dominate.
We study the energetic and dynamic stability of coreless vortices in nonrotated spin-1 Bose-Einstein condensates, trapped with a three-dimensional optical potential and a Ioffe-Pritchard field. The stability of stationary vortex states is investigated by solving the corresponding Bogoliubov equations. We show that the quasiparticle excitations corresponding to axisymmetric stationary states can be taken to be eigenstates of angular momentum in the axial direction. Our results show that coreless vortex states can occur as local or global minima of the condensate energy or become energetically or dynamically unstable depending on the parameters of the Ioffe-Pritchard field. The experimentally most relevant coreless vortex state containing a doubly quantized vortex in one of the hyperfine spin components turned out to have very nontrivial stability regions, and especially a quasiperiodic dynamic instability region which corresponds to splitting of the doubly quantized vortex.