Spontaneous Vortex Lattices in Quasi 2D Dipolar Spinor Condensates

📝 Abstract

Motivated by recent experiments\cite{BA}\cite{BB}, we study quasi 2D ferromagnetic condensates with various aspect ratios. We find that in zero magnetic field, dipolar energy generates a local energy minimum with all the spins lie in the 2D plane forming a row of {\em circular} spin textures with {\em alternating} orientation, corresponding to a packing of vortices of {\em identical} vorticity in different spin components. In a large magnetic field, the system can fall into a long lived dynamical state consisting of an array of elliptic and hyperbolic Mermin-Ho spin textures, while the true equilibrium is an uniaxial spin density wave with a single wave-vector along the magnetic field, and a wavelength similar to the characteristic length of the long lived vortex array state.

💡 Analysis

Motivated by recent experiments\cite{BA}\cite{BB}, we study quasi 2D ferromagnetic condensates with various aspect ratios. We find that in zero magnetic field, dipolar energy generates a local energy minimum with all the spins lie in the 2D plane forming a row of {\em circular} spin textures with {\em alternating} orientation, corresponding to a packing of vortices of {\em identical} vorticity in different spin components. In a large magnetic field, the system can fall into a long lived dynamical state consisting of an array of elliptic and hyperbolic Mermin-Ho spin textures, while the true equilibrium is an uniaxial spin density wave with a single wave-vector along the magnetic field, and a wavelength similar to the characteristic length of the long lived vortex array state.

📄 Content

Spontaneous Vortex Lattices in Quasi 2D Dipolar Spinor Condensates Jian Zhang†∗and Tin-Lun Ho† †Department of Physics, The Ohio State University, Columbus, OH 43210 ∗Institute of Advanced Study, Tsinghua University, Beijing 10086, China (Dated: August 3, 2021) Motivated by recent experiments[1][2], we study quasi 2D ferromagnetic condensates with various aspect ratios. We find that in zero magnetic field, dipolar energy generates a local energy minimum with all the spins lie in the 2D plane forming a row of circular spin textures with alternating orientation, corresponding to a packing of vortices of identical vorticity in different spin components. In a large magnetic field, the system can fall into a long lived dynamical state consisting of an array of elliptic and hyperbolic Mermin-Ho spin textures, while the true equilibrium is an uniaxial spin density wave with a single wave-vector along the magnetic field, and a wavelength similar to the characteristic length of the long lived vortex array state. The condensates of bosons with non-zero spins, known as spinor condensate, are remarkable superfluids. In ad- dition to broken gauge symmetry, they also have broken symmetries in spin space. The spin degrees of freedom lead to a variety ground states, which proliferates rapidly as the value of spin increases. Since different spin com- ponents can be mixed through spin rotation, there is considerable interplay between spin and gauge degrees of freedom, leading to a whole host of new macroscopic quantum phenomena. The simplest spinor condensates are those for spin- 1 bosons, such as the F = 1 hyperfine states of 23Na and 87Rb. The ground state of 23Na is a non-mangetic “polar” condensate whereas 87Rb is a ferromagnetic condensate[3]. In the case of ferromagnetic condensates, they possess an additional “spin-gauge” symmetry which makes non-uniform spin textures behave like vortices[4]. The system can respond to external rotation through spin deformation. Any attempt to bend the spin will also gen- erate vorticity. The magnetic nature of spinor condensates naturally leads to the consideration of dipolar energy, which is in- trinsic to alkali atoms. Since dipole energy can generate non-uniform spin textures, it will generate vorticity. In- deed, Yi and Pu have shown that a 87Rb condensate in a sufficiently flat cylindrical potential will form a circu- lar spin texture, which is a vortex of ferromagnetic con- densate with a polar core[5]. Recently, experiments at Berkeley have shown that a 87Rb condensate with a he- lical texture can decay into a random spin textures[6]. By estimating the energy of the final state, the authors suggest that the phenomenon is caused by dipolar en- ergy. More recently, the Berkeley group has found that a pancake like condensate of 87Rb can develop a texture with periodically modulated spin-spin correlation rotat- ing rapidly about an in-plane magnetic field[1, 2]. They suggest that this effect is also due to dipolar energy. Dipolar effects are highly geometry dependent. In this paper, we would like to point out that some key features of quasi 2D 87Rb condensate due to dipolar interactions. Much of what we discuss also apply to other ferromgnetic condensates. We shall consider an anisotropic trap with frequencies ωz » ωy > ωx, as in ref.[1, 2, 6]. The condensate is then a thin anisotropic slab in the xy-plane with Thomas-Fermi radii Rx, Ry such that Rx/Ry = (ωy/ωx) ≡λ, (λ ∼10 In ref.[1, 2, 6]). In our discussions, we choose the normal to the conden- sate slab, ˆz, to be the spin quantization axis for the con- densate wavefunction ΨT = (ψ1, ψ0, ψ−1), where the su- perscript “T” stands for transpose. We shall show that : (1) In zero magnetic field, dipolar energy leads to a lo- cal energy minimum consisting of a row of circular spin textures with alternating spin orientations in the long direction x, with all the spins in the xy-plane. These textures are of the Yi-Pu type[5], with a polar core and a size determined by the Thomas-Fermi radius in the short direction, Ry. This state amounts to an array of vortices in identical vorticity in the ψ1 component, with ψ−1 be- ing its time reversed partner. We have also found an analytic expression that well approximates this state. (2) In a large magnetic field B in the xy-plane, the spins rotate about ˆB and experience a time averaged dipolar energy. We find that this energy is very flat in spin space around a class of textures which is an array of elliptical and hyperbolic Mermin-Ho vortices. These states are not local minima. They will eventually evolve to the true minimum, which is an a uniaxial spin density wave (or “stripe phase” for short) with a single wave-vector along ˆB. Due to the flatness of the energy surface, the vortex lattice textures will be very long lived during dynamical evolutions. (A) Basic Structures: We first consider some spin textures relevant for later discussions. We shall write the condensate wavefunction as Ψµ(x) = p n(x)ζµ(x), where n

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