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arXiv:1204.3177 · Physical Review Letters

Stable Skyrmions in SU(2) Gauged Bose-Einstein Condensates

Takuto Kawakami, Takeshi Mizushima, Muneto Nitta, Kazushige Machida

Published Jul 2, 2012• 173 citations
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Abstract

We demonstrate that the three-dimensional Skyrmion, which has remained elusive so far, spontaneously appears as the ground state of SU(2) symmetric Bose-Einstein condensates coupled with a non-Abelian gauge field. The gauge field is a three-dimensional analogue of the Rashba spin-orbit coupling. Upon squashing the SO(3) symmetric gauge field to one- or two-dimensional shapes, we find that the ground state continuously undergoes a change from a three-dimensional to a one- or two-dimensional Skyrmion, which is identified by estimating winding numbers and helicity. All of the emerged Skyrmions are physically understandable with the concept of the helical modulation in a unified way. These topological objects might potentially be realizable in two-component Bose-Einstein condensates experimentally.

DOI
10.1103/physrevlett.109.015301

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We demonstrate that the three-dimensional Skyrmion, which has remained elusive so far, spontaneously appears as the ground state of SU(2) symmetric Bose-Einstein condensates coupled with a non-Abelian gauge field. The gauge field is a three-dimensional analogue of the Rashba spin-orbit coupling. Upon squashing the SO(3) symmetric gauge field to one- or two-dimensional shapes, we find that the ground state continuously undergoes a change from a three-dimensional to a one- or two-dimensional Skyrmion, which is identified by estimating winding numbers and helicity. All of the emerged Skyrmions are physically understandable with the concept of the helical modulation in a unified way. These topological objects might potentially be realizable in two-component Bose-Einstein condensates experimentally.
For the Bose–Einstein condensate in the non-Abelian gauge field, the atomic wavefunction obtains helical modulation, which would lead to nontrivial objects, such as three-dimensional Skyrmion. The spin–orbit coupled condensate is usually capable of supporting Skyrmion ground states, which can be well understood through the wavefunction modulation vector concept. In the present manuscript, the modulation vector is characterized both analytically and numerically with respect to the spin–orbit coupling strength. For the SU(2) symmetric condensate, the modulation vector of the wavefunction is linearly proportional to the spin–orbit coupling strength. In the presence of spin-dependent interaction, the modulus of the modulation vector is predicted to increase or decrease according to the sign of the interaction parameter, resulting in a shrunken or expanded Skyrmion. However, the Skyrmion is unstable for a certain area of the parameter space of the spin–orbit coupling and spin-dependent interaction. The condensate favors a single Skyrmion state or the Skyrmion lattice state according to the strength of the spin–orbit coupling.
We revisit ground states of spinor Bose-Einstein condensates with a Rashba spin-orbit coupling, and find that various types of vortices show up without rotation as a direct consequence of spontaneous symmetry breaking into a combined gauge, spin, and space rotation symmetry. The combined symmetry also determines the vortex-core spin state at the rotating center. For the continuous combined symmetry, the spin rotates by $2\ensuremath{\pi}$ around the rotating axis, whereas for the discrete combined symmetry, we further need 2F2F2F quantum numbers to determine how much the spin rotates for the spin-FFF system. For lattice phases we find that in the ground state the topological charge for each unit cell vanishes. However, we find two types of highly symmetric lattices with nontrivial topological charges for the spin-12\frac{1}{2}21​ system based on the symmetry classification, and show that they are Skyrmion crystals.
The recent experimental realization of spin-orbit coupling (SOC) in spin-1 ultracold atoms opens up an interesting avenue for exploring new quantum states and novel quantum phenomena in large-spin systems. Two types of two-dimensional double-pair vortices that are connected by a domain-wall can be accommodated in the antiferromagnetic spin-1 Bose–Einstein condensate (BEC). One type, named a 2D double-pair skyrmion, which differs from the conventional 2D skyrmion in the F = 1 polar BEC, hosts a pair of vortices and displays a meron-pair texture with a unit topological charge. The other type has a meron-pair texture but with null topological charge. These two types of double-pair vortices can be naturally generated from a vortex-free Gaussian wave packet by incorporating a non-Abelian gauge field into the spinor condensate.
Searching for novel topological objects is always an intriguing task for scientists in various fields. We study a three-dimensional (3D) topological structure called a 3D dimeron in trapped two-component Bose-Einstein condensates. The 3D dimeron differs from the conventional 3D skyrmion for the condensates hosting two interlocked vortex rings. We demonstrate that the vortex rings are connected by a singular string and the complexity constitutes a vortex molecule. The stability of the 3D dimeron is examined in two different models using the imaginary time evolution method. We find that the stable 3D dimeron can be naturally generated from a vortex-free Gaussian wave packet incorporating a synthetic non-Abelian gauge potential into the condensates.
Motivated by recent studies of Bose Einstein condensates (BECs) under synthetic gauge fields, we study the textures of F = 2 BECs with a Rashba like spin-orbit coupling (SOC). By analyzing the SOC term, we demonstrate that the SOC favors a helical modulation of order parameters (OPs). In addition, we find the stable textures obtained by the numerical minimization of the Gross-Pitaevskii energy functional and interpret them as a one- or two-dimensional helical modulation. In the parameter region where the polar OP is favored, the emergence of the skyrmion of the uniaxial polar OP and its energetics relative to one-dimensional helical modulation of biaxial polar OP are discussed in details. Through the comparison of the lattice texture in the cyclic favored parameter region and uniaxial polar skyrmion, we show the instability of the lattice of the uniaxial polar skyrmion.
Attractive Bose-Einstein condensates of dilute alkali gases are unstable against collapse in two- and three-dimensional free space. Nevertheless, we demonstrate that the spin-orbit coupling of the two-component condensates counteracts the tendency of collapse and makes the system preferable to an extended spatial distribution in the three-dimensional case. Furthermore, stable topological objects can be formed in the condensates, which are shown to be the lowest energy states. Two configurations of the density profiles, called three-dimensional skyrmion and three-dimensional dimeron, respectively, are identified depending on the strength of the spin-orbit coupling.