Creation of skyrmions in a spinor Bose-Einstein condensate

Abstract

We propose a scheme for the creation of skyrmions (coreless vortices) in a Bose-Einstein condensate with hyperfine spin F = 1. In this scheme, four traveling-wave laser beams, with Gaussian or Laguerre-Gaussian transverse profiles, induce Raman transitions with an anomalous dependence on the laser polarization, thereby generating the optical potential required for producing skyrmions. 03.75.Fi, The recent experimental success in all-optical trapping of an atomic Bose-Einstein condensate (BEC) [1] opens the prospect of studies into the internal structure of spinor BECs, including the possibility of creating vortex states without core, or skyrmions, in the BEC [2][3][4][5][6]. Skyrmions, which do not have an ordinary vortex core due to the spin degree of freedom, offer a myriad of new physical phenomena beyond those presented by other vortex states. One feature is the reduction of kinetic energy associated with the rotation by transferring this energy to the spin. Another interesting property of skyrmions is that they do not represent a topological excitation, although their flux is vortex-like far away from the line of symmetry [7]. In this paper, we propose an optical method to create skyrmions in a condensate of alkali atoms. This method, which employs laser beams and Raman transition to generate skyrmions in the BEC, is related to proposals for the creation of vortices in a single-component BEC [8,9].For a BEC with hyperfine spin F = 1, a skyrmion wave function, which is axially symmetric around the z-axis, has the form [2]with ρ(z, r ⊥ ) the total density of atoms, r ⊥ = (x, y) the transverse coordinate vector and r ⊥ = |r ⊥ | the radial distance from the z-axis. The angle φ corresponds to the orientation of r ⊥ in the x-y-plane. The function β(r ⊥ ), which characterizes the spin state of the condensed atoms, is related to the superfluid velocity v s of the system bywhere e φ ≡ (cos φ, − sin φ, 0) is the unit vector in the φ direction. Because a skyrmion has no vortex core, β(0) = 0 must hold in order to avoid a singularity. For β(r ⊥ ) = π/2, the superfluid velocity reduces to that of an ordinary vortex state.To create a skyrmion we consider a BEC which first is magnetically trapped in the m = −1 hyperfine state. We assume that the trap is then switched off and the optical potential is applied to the BEC. Thus, our initial state is given by ψ −1 = ρ(r ⊥ , z) and ψ 0 = ψ 1 = 0. Our objective is to design an optical potential which transfers this state into the state (1).Generally a coherent optical potential for atoms is created by applying several highly detuned laser beams which induce Raman transitions between different angular momentum states or internal states of the atomic BEC (cf Fig. 1). In order to preserve the cylindrical symmetry of the skyrmions along the z-axis, the laser beams must also propagate along this axis. This restriction raises the following challenge: the polarization orientation of laser beams propagating along the z-axis is generally presumed to be in a superposition of e x and e y ....

Mermin-Ho Vortex in Ferromagnetic Spinor Bose-Einstein Condensates

“…Khawaja and Stoof[10] study a skyrmion in the ferromagnetic BEC with F = 1 and conclude that the skyrmion is not a thermodynamically stable object without rotation. Once created, its radius shrinks to vanish in spite of ingenious proposals [11,12,13,14] as to how to created it and to detect it.Here we demonstrate that in the ferromagnetic spinor BEC with F = 1 trapped in a harmonic potential the Mermin-Ho vortex and Anderson-Toulouse vortex are thermodynamically stable for a certain region of the external rotation frequency and the magnetization of the system. These vortices are stabler than the singular vortices and other possible non-axis symmetric vortices.…”

“…There is a rich variety of topological defect structures, which are already predicted in the earlier studies [6,7] on the spinor BEC. These are followed by others [8,9,10,11,12,13,14,15,16] who examine these topological structures more closely, such as skyrmion, monopole, meron or axis-symmetric or non axis-symmetric vortices both for antiferromagnetic and ferromagnetic cases.Superfluid 3 He is analogous to the spinor BEC where the neutral Cooper pair possesses the orbital and spin degrees of freedom, thus the order parameter is a multi-component [17]. Among various topological structures, the Mermin-Ho (MH) [18] and Anderson-Toulouse (AT) [19] vortices of a coreless and non-singular l-vector texture are proposed in 3 He-A phase.…”

“…There is a rich variety of topological defect structures, which are already predicted in the earlier studies [6,7] on the spinor BEC. These are followed by others [8,9,10,11,12,13,14,15,16] who examine these topological structures more closely, such as skyrmion, monopole, meron or axis-symmetric or non axis-symmetric vortices both for antiferromagnetic and ferromagnetic cases.…”

Raman coupling of Zeeman sublevels in an alkali-metal Bose-Einstein condensate

“…By using beam modes with polarization components in the direction of propagation, this can be accomplished while still using a collinear beam geometry, i.e. by driving anomalous Raman transitions [36]. Experimental work exploring this intriguing possibility is in progress.…”

Kinetic Thomas–Fermi solutions of the Gross–Pitaevskii equation

“…The excitation of each momentum component requires a pair of counterpropagating laser beams with frequencies ω and ω + Ω, i.e., in total eight beams are necessary forming a pair of crossed bichromatic standing waves. Similar multi-beam variants of SRS have been proposed as a means to excite vortex or skyrmion states in a BEC [17,18].…”