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arXiv:2012.00872 · Physical Review A

Multiring, stripe, and superlattice solitons in a spin-orbit-coupled spin-1 condensate

Sadhan K. Adhikari

Published Jan 7, 2021• 36 citations
arXivHTMLPDF

Abstract

We demonstrate exotic stable quasi-two-dimensional solitons in a Rashba or a Dresselhaus spin-orbit (SO)-coupled hyperfine spin-1 ($F=1$) trapless antiferromagnetic Bose-Einstein condensate using the mean-field Gross-Pitaevskii equation. For weak SO coupling, the solitons are of the $(\ensuremath{-}1,0,+1)$ or $(+1,0,\ensuremath{-}1)$ type with intrinsic vorticity, for Rashba or Dresselhaus SO coupling, where the numbers in the parentheses denote the angular momentum in the spin components ${F}_{z}=+1,0,\ensuremath{-}1$, respectively. For intermediate SO coupling, the solitons have a multiring structure maintaining the above-mentioned vortices in the respective components. For larger SO coupling, superlattice solitons with a square-lattice structure in total density are found in addition to stripe solitons with a stripe pattern in the component densities with no periodic modulation in the total density.

DOI
10.1103/physreva.103.l011301

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We demonstrate exotic stable quasi-two-dimensional solitons in a Rashba or a Dresselhaus spin-orbit (SO)-coupled hyperfine spin-1 (F=1F=1F=1) trapless antiferromagnetic Bose-Einstein condensate using the mean-field Gross-Pitaevskii equation. For weak SO coupling, the solitons are of the (\ensuremath−1,0,+1)(\ensuremath{-}1,0,+1)(\ensuremath−1,0,+1) or (+1,0,\ensuremath−1)(+1,0,\ensuremath{-}1)(+1,0,\ensuremath−1) type with intrinsic vorticity, for Rashba or Dresselhaus SO coupling, where the numbers in the parentheses denote the angular momentum in the spin components Fz=+1,0,\ensuremath−1{F}_{z}=+1,0,\ensuremath{-}1Fz​=+1,0,\ensuremath−1, respectively. For intermediate SO coupling, the solitons have a multiring structure maintaining the above-mentioned vortices in the respective components. For larger SO coupling, superlattice solitons with a square-lattice structure in total density are found in addition to stripe solitons with a stripe pattern in the component densities with no periodic modulation in the total density.
Abstract We demonstrate the formation of stable spatially-ordered states in a uniform and also trapped quasi-two-dimensional (quasi-2D) Rashba or Dresselhaus spin–orbit (SO) coupled pseudo spin-1/2 Bose–Einstein condensate using the mean-field Gross–Pitaevskii equation. For weak SO coupling, one can have a circularly-symmetric (0, +1)- or (0, −1)-type multi-ring state with intrinsic vorticity, for Rashba or Dresselhaus SO coupling, respectively, where the numbers in the parentheses denote the net angular momentum projection in the two components, in addition to a circularly-asymmetric degenerate state with zero net angular momentum projection. For intermediate SO couplings, in addition to the above two types, one can also have states with stripe pattern in component densities with no periodic modulation in total density. The stripe state continues to exist for large SO coupling. In addition, a new spatially-periodic state appears in the uniform system: a superlattice state, possessing some properties of a supersolid , with a square-lattice pattern in component densities and also in total density. In a trapped system the superlattice state is slightly different with multi-ring pattern in component density and a square-lattice pattern in total density. For an equal mixture of Rashba and Dresselhaus SO couplings, in both uniform and trapped systems, only stripe states are found for all strengths of SO couplings. In a uniform system all these states are quasi-2D solitonic states.
We study the formation of spin-1 symbiotic spinor solitons in a quasi-one- (quasi-1D) and quasi-two-dimensional (quasi-2D) hyperfine spin F=1 ferromagnetic Bose-Einstein condensate (BEC). The symbiotic solitons necessarily have a repulsive intraspecies interaction and are bound due to an attractive interspecies interaction. Due to a collapse instability in higher dimensions, an additional spin-orbit coupling is necessary to stabilize a quasi-2D symbiotic spinor soliton. Although a quasi-1D symbiotic soliton has a simple Gaussian-type density distribution, novel spatial periodic structure in density is found in quasi-2D symbiotic SO-coupled spinor solitons. For a weak SO coupling, the quasi-2D solitons are of the (-1,0,+1) or (+1,0,-1) type with intrinsic vorticity and multiring structure, for Rashba or Dresselhaus SO coupling, respectively, where the numbers in the parentheses are angular momenta projections in spin components F_{z}=+1,0,-1, respectively. For a strong SO coupling, stripe and superlattice solitons, respectively, with a stripe and square-lattice modulation in density, are found in addition to the multiring solitons. The stationary states were obtained by imaginary-time propagation of a mean-field model; dynamical stability of the solitons was established by real-time propagation over a long period of time. The possibility of the creation of such a soliton by removing the trap of a confined spin-1 BEC in a laboratory is also demonstrated.
We study the formation of solitons in a uniform quasi-two-dimensional (quasi-2D) spin-orbit (SO) coupled self-repulsive binary dipolar and nondipolar Bose-Einstein condensate (BEC) using the mean-field Gross-Pitaevskii equation. For a weak SO coupling, in a nondipolar BEC, one can have three types of degenerate solitons: a multiring soliton with intrinsic vorticity of angular momentum projection +1 or -1 in one component and 0 in the other, a circularly asymmetric soliton, and a stripe soliton with stripes in the density. For an intermediate SO coupling, the multiring soliton ceases to exist and a square-lattice soliton appears with a spatially periodic pattern in density on a square lattice, in addition to the degenerate circularly asymmetric and stripe solitons. In the presence of a dipolar interaction, with the polarization direction aligned in the quasi-2D plane, only the degenerate circularly asymmetric and stripe solitons appear.
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 supersolid-like crystalline structures emerging in the stationary states of a quasi-two-dimensional spin-orbit (SO)-coupled spin-2 condensate in the ferromagnetic, cyclic, and antiferromagnetic phases by solving a mean-field model. Interplay of different strengths of SO coupling and interatomic interactions gives rise to a variety of nontrivial density patterns in the emergent solutions. For small SO-coupling strengths $\ensuremath{\gamma}$ $(\ensuremath{\gamma}\ensuremath{\approx}0.5)$, the ground state is an axisymmetric multiring soliton for polar, cyclic, and weakly ferromagnetic interactions, whereas for stronger ferromagnetic interactions a circularly asymmetric soliton emerges as the ground state. Depending on the values of interaction parameters, with an increase in SO-coupling strength, a stripe phase may also emerge as the ground state for polar and cyclic interactions. For intermediate values of SO-coupling strength $(\ensuremath{\gamma}\ensuremath{\approx}1)$, in addition to these solitons, one could have a quasidegenerate triangular-lattice soliton in all magnetic phases. On further increasing the SO-coupling strength $(\ensuremath{\gamma}⪆4)$, a square-lattice and a superstripe soliton emerge as quasidegenerate states. The emergence of all these solitons can be inferred from a study of solutions of the single-particle Hamiltonian.
Recently, research of solitons in Bose–Einstein condensates has become a popular topic. Here, we mainly study exact analytical solutions of Gross–Pitaevskii equations describing spin-orbit coupled spin-1 Bose–Einstein condensates. To begin with, we show the analytical relation between different types of one-dimensional spin-orbit coupling and Zeeman effect. In addition, we find a transformation that can simplify the three-component Gross–Pitaevskii equations with spin-orbit coupling into the nonlinear Schrödinger equation. The abundant stripe phase and dynamic characteristics of the system are investigated.