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Physical Review A

Topological defects and inhomogeneous spin patterns induced by the quadratic Zeeman effect in spin-1 Bose-Einstein condensates

Dun Zhao, Shu-Wei Song, Lin Wen, Zai-Dong Li, Hong‐Gang Luo, Wu-Ming Liu

Published Jan 20, 2015• 16 citations
Publisher

Abstract

We obtain analytically two kinds of inhomogeneous spin domain configurations endowed with time-dependent periodic domain walls in spin-1 Bose-Einstein condensate, which result from the positive and negative quadratic Zeeman effects, respectively. It is shown analytically that the topological defects are given by the unmagnetized atoms, at which the order parameter is of polar phase, but besides them, the order parameter is of axisymmetry-broken phase. The quadratic Zeeman effect can be tuned to induce the dynamical phase transition of the spin domain, and its sign can affect the topological structure of the spin pattern. These features arise from the axisymmetry breaking due to the pointwise-different population exchange between the sublevels determined uniquely by the quadratic Zeeman effects. The related experimental observations for the spin-1 $^{87}\mathrm{Rb}$ and $^{23}\mathrm{Na}$ condensates are also discussed.

DOI
10.1103/physreva.91.013619

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We obtain analytically two kinds of inhomogeneous spin domain configurations endowed with time-dependent periodic domain walls in spin-1 Bose-Einstein condensate, which result from the positive and negative quadratic Zeeman effects, respectively. It is shown analytically that the topological defects are given by the unmagnetized atoms, at which the order parameter is of polar phase, but besides them, the order parameter is of axisymmetry-broken phase. The quadratic Zeeman effect can be tuned to induce the dynamical phase transition of the spin domain, and its sign can affect the topological structure of the spin pattern. These features arise from the axisymmetry breaking due to the pointwise-different population exchange between the sublevels determined uniquely by the quadratic Zeeman effects. The related experimental observations for the spin-1 87Rb^{87}\mathrm{Rb}87Rb and 23Na^{23}\mathrm{Na}23Na condensates are also discussed.
We show two kinds of inhomogeneous spin domain possessing Néel-like domain walls in spin-1 Bose-Einstein condensate, which are induced by the positive and negative quadratic Zeeman effect (QZE) respectively. In both cases, the spin density distribution is inhomogeneous and has zeros where the magnetization vanishes. For positive and negative QZE, the spin patterns and topological structures are remarkably different. Such phenomena are due to the pointwise different axisymmetry-breaking caused by the pointwise different population exchange between the sublevels, arising uniquely from the QZE. We analyze in detail the inhomogeneous domain formation and related experimental observations for the spin-1 87^{87}87Rb and 23^{23}23Na condensate.
We study the dynamics of the quantum phase transition of a ferromagnetic spin-1 Bose-Einstein condensate (BEC) from the polar phase to the broken-axisymmetry phase by changing magnetic field, and find the spontaneous formation of spinor domain walls followed by the creation of polar-core spin vortices. We also find that the spin textures depend very sensitively on the initial noise distribution, and that an anisotropic and colored initial noise is needed to reproduce the Berkeley experiment [Sadler et al., Nature (London) 443, 312 (2006)]. The dynamics of vortex nucleation and the number of created vortices depend also on the manner in which the magnetic field is changed. We point out an analogy between the formation of spin vortices from domain walls in a spinor BEC and that of vortex-antivortex pairs from dark solitons in a scalar BEC.
We investigate the effects of thermal and quantum fluctuations on the phase diagram of a spin-1 87Rb{}^{\text{87}}\text{Rb}87Rb Bose-Einstein condensate (BEC) under the quadratic Zeeman effect. Due to the large ratio of spin-independent to spin-dependent interactions of 87Rb{}^{\text{87}}\text{Rb}87Rb atoms, the effect of noncondensed atoms on the condensate is much more significant than that in scalar BECs. We find that the condensate and spontaneous magnetization emerge at different temperatures when the ground state is in the broken-axisymmetry phase. In this phase, a magnetized condensate induces spin coherence of noncondensed atoms in different magnetic sublevels, resulting in temperature-dependent magnetization of the noncondensate. We also examine the effect of quantum fluctuations on the order parameter at absolute zero and find that the ground-state phase diagram is significantly altered by quantum depletion.
We find a novel topological defect in a spin-nematic superfluid theoretically. A quantized vortex spontaneously breaks its axisymmetry, leading to an elliptic vortex in nematic-spin Bose-Einstein condensates with small positive quadratic Zeeman effect. The new vortex is considered the Joukowski transform of a conventional vortex. Its oblateness grows when the Zeeman length exceeds the spin healing length. This structure is sustained by balancing the hydrodynamic potential and the elasticity of a soliton connecting two spin spots, which are observable by in situ magnetization imaging. The theoretical analysis clearly defines the difference between half quantum vortices of the polar and antiferromagnetic phases in spin-1 condensates.
We consider coherent states of weakly interacting bosons under the conditions of external resonant excitation, with a focus on a two-dimensional polariton fluid driven by a plane electromagnetic wave near the ground state. The coherent driving breaks the U(1) symmetry explicitly, which prevents the occurrence of quantum vortices in a uniform scalar condensate. Surprisingly, a spinor (two-component) system of the same kind admits topological excitations, such as domain walls of relative phase or confined half-vortex molecules, typical of a freely evolving spinor Bose system. Opposite-phase domains arise from the spontaneous breakdown of the spin symmetry (Z2)(\mathbb{Z}_2)(Z2​). Domain walls form with time even when the initial state of the system is uniform or completely disordered; they fall into different topological types distinguished by the total phase variation in the transverse direction. One type of domain walls is similar to ``magnetic'' solitons in Bose-Einstein condensates and exhibits nonzero spin polarization whose sign depends on the direction of motion. Domain walls of the second type, by contrast, behave like monopoles with broken spatiotemporal symmetry and tend to move in certain preferred directions. The interaction of vortices and domain walls results in a long-range ordering of the system.