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arXiv:1701.04591 · Physical Review A

Vortex pairs in a spin-orbit-coupled Bose-Einstein condensate

Masaya Kato, Xiao-Fei Zhang, Hiroki Saito

Published Apr 5, 2017• 33 citations
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Abstract

Static and dynamic properties of vortices in a two-component Bose-Einstein condensate with Rashba spin-orbit coupling are investigated. The mass current around a vortex core in the plane-wave phase is found to be deformed by the spin-orbit coupling, and this makes the dynamics of the vortex pairs quite different from those in a scalar Bose-Einstein condensate. The velocity of a vortex-antivortex pair is much smaller than that without spin-orbit coupling, and there exist stationary states. Two vortices with the same circulation move away from each other or unite to form a stationary state.

DOI
10.1103/physreva.95.043605

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Static and dynamic properties of vortices in a two-component Bose-Einstein condensate with Rashba spin-orbit coupling are investigated. The mass current around a vortex core in the plane-wave phase is found to be deformed by the spin-orbit coupling, and this makes the dynamics of the vortex pairs quite different from those in a scalar Bose-Einstein condensate. The velocity of a vortex-antivortex pair is much smaller than that without spin-orbit coupling, and there exist stationary states. Two vortices with the same circulation move away from each other or unite to form a stationary state.
Three types of vortex-pair are identified in two-component Bose–Einstein condensates (BEC) of different kinds of spin-orbit coupling. One type holds the two vortices in one component of the twocomponent condensates. Both the other two types hold a vortex in each component of the twocomponent condensates, and exhibit meron-pair textures that have either null or unit topological charge, respectively. The cores of the two vortices are connected by a string of the relative phase jump. These vortex pairs can be generated from a vortex-free wave packet by incorporating different non- Abelian gauge field into the BEC. When a Rabi coupling is introduced, the distance between the two cores is effectively controlled by the Rabi coupling strength and a transition of vortex configurations is observed.
I use a time-dependent Lagrangian formalism and a variational trial function to study the dynamics of a two-component vortex in a spin-orbit coupled Bose-Einstein condensate (BEC). For a single-component BEC, various experiments have validated this theoretical approach, for example a thermal quench that yields a quantized vortex in roughly 25% of trials. To be definite, I assume the specific spin-orbit form used in recent NIST experiments, which introduces a spatial asymmetry because of the external Raman laser beams. I here generalize this formalism to include a two-component order parameter that has quantized circulation in each component but not necessarily with the same circulation. For example a singly quantized vortex in just one component yields a BEC analog of the half-quantum vortex familiar in 3^33He-A and in ppp-wave chiral superconductors. This and other unusual two-component vortices have both periodic trajectories and unbounded trajectories that leave the condensate, depending on the initial conditions. The optimized phase of the order parameter induces a term in the particle current that cancels the contribution from the vector potential, leaving pure circulating current around the vortex.
We analyze the dynamics of Josephson vortex states in two-component Bose-Einstein condensates with Rashba-Dresselhaus spin-orbit coupling by using the Gross-Pitaevskii equation. In one dimension, both in homogeneous and harmonically trapped systems, we report on stationary states containing doubly charged, static Josephson vortices. In multidimensional systems, we find stable Josephson vortices in a regime of parameters typical of current experiments with 87Rb^{87}\text{Rb}87Rb atoms. In addition, we discuss the instability regime of Josephson vortices in disk-shaped condensates, where the snake instability operates and vortex dipoles emerge. We study the rich dynamics that they exhibit in different regimes of the spin-orbit-coupled condensate depending on the orientation of the Josephson vortices.
We study the vortex-lattice formation in a rotating Rashba spin-orbit (SO) coupled quasi-two-dimensional (quasi-2D) hyper-fine spin-1 spinor Bose-Einstein condensate (BEC) in the x-y plane using a numerical solution of the underlying mean-field Gross-Pitaevskii equation. In this case, the non-rotating Rashba SO-coupled spinor BEC can have topological excitation in the form of vortices of different angular momenta in the three components, e.g. the (0, +1, +2)- and (-1, 0, +1)-type states in ferromagnetic and anti-ferromagnetic spinor BEC: the numbers in the parenthesis denote the intrinsic angular momentum of the vortex states of the three components with the negative sign denoting an anti-vortex. The presence of these states with intrinsic vorticity breaks the symmetry between rotation with vorticity along the z and -z axes and thus generates a rich variety of vortex-lattice and anti-vortex-lattice states in a rotating quasi-2D spin-1 spinor ferromagnetic and anti-ferromagnetic BEC, not possible in a scalar BEC. For weak SO coupling, we find two types of symmetries of these states - hexagonal and 'square'. The hexagonal (square) symmetry state has vortices arranged in closed concentric orbits with a maximum of 6, 12, 18… (8, 12, 16…) vortices in successive orbits. Of these two symmetries, the square vortex-lattice state is found to have the smaller energy.
We study vortex dynamics in trapped two-component Bose-Einstein condensates with a laser-induced spin-orbit coupling using the numerical analysis of the Gross-Pitaevskii equation. The spin-orbit coupling leads to three distinct ground-state phases, which depend on some experimentally controllable parameters. When a vortex is put in one or both of the two-component condensates, the vortex dynamics exhibits very different behaviors in each phase, which can be observed in experiments. These dynamical behaviors can be understood by clarifying the stable vortex structure realized in each phase.
The dynamics of pseudo-spin-1/2 Bose-Einstein condensates with weak spin-orbit coupling through a moving obstacle potential are studied numerically. Four types of wakes are observed and the phase diagrams are determined for different spin-orbit coupling strengths. The conditions to form Bénard-von Kármán vortex street are rather rigorous, and we investigate in detail the dynamical characteristics of the vortex streets. The two point vortices in a pair rotate around their center, and the angular velocity and their distance oscillate periodically. The oscillation intensifies with increasing spin-orbit coupling strengths, and it makes part of the vortex pairs dissociate into separate vortices or combine into single ones and destroys the vortex street in the end. The width b of the street and the distance l between two consecutive vortex pairs of the same circulation are determined by the potential radius and its moving velocity, respectively. The b/l ratios are independent of the spin-orbit coupling strength and fall in the range 0.19-0.27, which is a little smaller than the stability criterion 0.28 for classical fluids. Proper b/l ratios are necessary to form Bénard-von Kármán vortex street, but the spin-orbit coupling strength affects the stability of the street patterns. Finally, we propose a protocol to experimentally realize the vortex street in ^{87}Rb spin-orbit-coupling Bose-Einstein condensates.