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We investigate the collapse dynamics of a dipolar condensate of 52Cr atoms when the s-wave scattering length characterizing the contact interaction is reduced below a critical value. A complex dynamics, involving an anisotropic, d-wave symmetric explosion of the condensate, is observed. The atom number decreases abruptly during the collapse. We find good agreement between our experimental results and those of a numerical simulation of the three-dimensional Gross-Pitaevskii equation, including contact and dipolar interactions as well as three-body losses. The simulation indicates that the collapse induces the formation of two vortex rings with opposite circulations.

The general properties of the order parameter of a dipolar spinor Bose-Einstein condensate are discussed based on the symmetries of interactions. An initially spin-polarized dipolar condensate is shown to dynamically generate a non-singular vortex via spin-orbit interactions, a phenomenon reminiscent of the Einstein-de Haas effect in ferromagnets.PACS numbers: 03.75. Mn,03.75.Nt,03.75.Kk,03.75.Lm The realization of a Bose-Einstein condensate (BEC) of 52 Cr [1, 2] marks a major development in degenerate quantum gases in that the interparticle interaction via magnetic dipoles in this BEC is much larger than those in other spinor BECs of alkali atoms. The long-range nature and anisotropy of the dipolar interaction pose challenging questions concerning the stability and superfluidity of the BEC [3,4,5,6,7,8,9,10]. The ground state of the 52 Cr atom has a total electronic spin of three and a nuclear spin of zero, and therefore the 52 Cr BEC has seven internal degrees of freedom. The interplay between dipolar and spinor interactions makes the order parameter of this system highly non-trivial [11,12,13]. Moreover, the dipole interaction couples the spin and orbital angular momenta so that an initial magnetization of the system causes the gas to rotate mechanically (Einstein-de Haas effect [14]) or, conversely, solid-body rotation of the system leads to its magnetization (Barnett effect [15]).This Letter investigates the Einstein-de Haas and Barnett effects in a spin-3 BEC system. We discuss the symmetry of the order parameter of a dipolar spinor BEC and study the dynamic formation of spin textures using numerical simulations of the seven-component nonlocal mean-field theory, which takes into account short-range (van der Waals) interactions and magnetic dipole-dipole interactions subject to a trapping potential and an external magnetic field.We first consider general properties of the order parameter by discussing two fundamental symmetries of the dipolar interaction between the magnetic dipole moments µ 1 = gµ Bŝ1 and µ 2 = gµ Bŝ2 , where g is the electron gfactor, µ B is the Bohr magneton, andŝ 1 andŝ 2 are the spin operators. The interaction between the magnetic dipoles located at r 1 and r 2 is described bŷwhere r 12 ≡ r 1 − r 2 , e 12 ≡ r 12 /r 12 , and c dd = µ 0 (gµ B ) 2 /4π with µ 0 being the magnetic permeability of the vacuum. The dipole interaction is invariant under simultaneous rotation in spin and coordinate spaces about an arbitrary axis, say the z-axis, so that the projected total angular momentum operatorŜ z +L z on that axis commutes withv dd , whereŜ =ŝ 1 +ŝ 2 is the total spin operator andL is the relative orbital angular momentum operator. Another symmetry of the dipolar interaction is the invariance under the transformation P z e −iπŜz , where P z : (x, y, z) → (x, y, −z) and. Thus, the eigenvalues of the following operators are conserved by the dipole interaction:A crucial observation is that these operators also commute with the short-range interactions. Thus if the confining potential is axi...

We consider the splitting mechanism of a multiply charged vortex into singly charged vortices in a Bose-Einstein condensate confined in a harmonic potential at zero temperature. The Bogoliubov equations support unstable modes with complex eigenfrequencies (CE modes), which cause the splitting instability without the influence of thermal atoms. The investigation of the excitation spectra shows that the negative-energy (NE) mode plays an important role in the appearance of the CE modes. The configuration of vortices in splitting is determined by the angular momentum of the associated NE mode. This structure has also been confirmed by the numerical simulation of the time-dependent Gross-Pitaevskii equation.ized order parameter = a HO / ͱ N, where a HO is the har-*Electronic address: yuki@scphys.kyoto-u.ac.jp PHYSICAL REVIEW A 70, 043610 (2004)

The spin vortices are shown to be created through the Kibble-Zurek (KZ) mechanism in a quantum phase transition of a spin-1 ferromagnetic Bose-Einstein condensate, when the applied magnetic field is quenched below a critical value. It is shown that the magnetic correlation functions have finite correlation lengths, and magnetizations at widely separated positions grow in random directions, resulting in spin vortices. We numerically confirm the scaling law that the winding number of spin vortices is proportional to the square root of the length of the closed path, and for slow quench, proportional to τ −1/6 Q with τQ being the quench time. The relation between the spin conservation and the KZ mechanism is discussed.

We show that knots of spin textures can be created in the polar phase of a spin-1 Bose-Einstein condensate, and discuss experimental schemes for their generation and probe, together with their lifetime.

We report on a study of the spin-1 ferromagnetic Bose-Einstein condensate with magnetic dipoledipole interactions. By solving the non-local Gross-Pitaevskii equations for this system, we find three ground-state phases. Moreover, we show that a substantial orbital angular momentum accompanied by chiral symmetry breaking emerges spontaneously in a certain parameter regime. We predict that all these phases can be observed in the spin-1 87 Rb condensate by changing the number of atoms or the trap frequency.PACS numbers: 03.75. Kk,03.75.Lm,03.75.Mn,03.75.Nt The magnetic dipole-dipole interaction in ferromagnets is responsible for a rich variety of spin structures [1]. Similar spin textures can also be expected to occur, due to the dipolar interaction, in ferromagnetic Bose-Einstein condensates (BECs), such as the spin-1 87 Rb BEC. However, spinor BECs differ from ferromagnets in that they exhibit spin-gauge symmetry that can generate mass flow by developing spin textures [2]. One may therefore wonder whether the dipole-induced spin texture can yield spontaneous mass current in the ground state. In this Letter we show that this is the case.An additional motivation for our work is the recent observation of a dipolar BEC in a system of spin-polarized 52 Cr atoms [3]. The ground state of spinor dipolar BECs has been studied by several researchers using a singlemode approximation [4,5,6]. Magnetism of dipolar BECs in one and two dimensional optical lattices has also been investigated [7,8]. In this Letter, we study the ground-state spin textures of a spinor dipolar BEC without invoking the single-mode approximation. We show the existence of three ground-state phases at zero magnetic field. In particular, we identify chiral spin-vortex phase and show that spontaneous circulation with broken chiral symmetry emerges in this phase.We consider a system of N spin-1 atoms with mass M confined in a spin-independent potential U trap (r) = M ω 2 (x 2 + y 2 + z 2 )/2. The Hamiltonian of the system is given by [9]whereψ m (r) is the annihilation operator of an atom in the magnetic sublevel m = 0, ±1 at point r,, F mm ′ represents the spin-1 matrix, andĤ dd is the contribution of the magnetic dipole-dipole interaction. The spin-independent and spin-dependent interactions are characterized by g 0 = 4π2 (a 0 + 2a 2 )/(3M ) and g 1 = 4π 2 (a 2 − a 0 )/(3M ), respectively, with a s (s = 0, 2) being the s-wave scattering length for the scattering channel with total spin s. For spin-1 87 Rb atoms we have 0 < −g 1 ≪ g 0 , and the ground state is ferromagnetic.The magnetic dipole moment of an atom is given bŷ µ = −g S µ BŜ + g I µ NÎ , where µ B is the Bohr magneton, µ N is the nuclear magneton,Ŝ andÎ are the electronic and nuclear spin angular momenta, respectively, and g S and g I are the Landé g factors. The matrix element of the dipole moment between magnetic sublevels in the spin-1 hyperfine manifold is shown to be m|μ|m ′ = g F µ B F mm ′ . For the case of 87 Rb with S = 1/2 and I = 3/2, we have g F = (g S + 5g I µ N /µ B )/4 ≃ 1/2. ...

We investigate the collision dynamics of two non-Abelian vortices and find that, unlike Abelian vortices, they neither reconnect themselves nor pass through each other, but create a rung between them in a topologically stable manner. Our predictions are verified using the model of the cyclic phase of a spin-2 spinor Bose-Einstein condensate.PACS numbers: 03.75.Mn, 05.30.Jp, 11.27.+d Quantized vortices are topological defects in the superfluid order-parameter, and their character depends on the topology of the order-parameter manifold of the system. In the case of single component Bose-Einstein Condensates (BECs), for example, the order-parameter manifold is U (1) and the quantized vortices are characterized by an additive group of integers. However, the situation is dramatically different for spinor BECs, where some phases accommodate non-Abelian vortices where the collision dynamics shows markedly different behavior from that of Abelian vortices. In this Letter, we investigate the collisional properties of non-Abelian vortices and their unique topological properties. In particular, we find that a rung vortex that bridges the colliding vortices is always formed upon collision, regardless of the ranges of the kinematic parameters. We verify these ideas with numerical simulations for a spin-2 BEC.The topological charge of a vortex can be identified by studying how the order-parameter changes along a closed path encircling the vortex. For U (1) vortices, the U (1) phase changes around the vortex by an integer multiple of 2π, and the topological charge is expressed by integers. In spinor BECs, on the other hand, the change of the orderparameter around the vortex involves not only the U (1) phase but also an SO(3) rotation of the spin. Therefore, there are phases in which topological charges of vortices do not commute with each other and form a non-Abelian group. We define such vortices with noncommutative topological charges as non-Abelian vortices [1].A salient feature of non-Abelian vortices manifests itself in the collision dynamics. In Abelian vortices, the following three types of collisions are possible: reconnection, passing through, and a formation of a rung vortex that bridges the colliding vortices. The reconnection of Abelian U (1) vortices has been studied theoretically [2], and observed recently in superfluid 4 He [3]. Moreover, there are some theoretical works which predict the rung structure that connects two attracting U (1) vortices [4] or U (1) × U (1) vortices [5].When the vortices are non-Abelian, the situation changes dramatically. It was predicted that the collision of two non-Abelian vortices produces a tangled and con- nected rung vortex [6]. In fact, for the case of two vortices with noncommutative topological charges, reconnection and passing through are topologically forbidden and only the formation of a rung vortex is allowed. This can be understood by considering algebraic and geometric structures of the vortices as shown in Fig 1. Let us consider two colliding vortices with topologi...