Ground, Symmetric and Central Vortex States in Rotating Bose-Einstein Condensates

Bao, Weizhu; Wang, Hanquan; Markowich, Peter A.

doi:10.4310/cms.2005.v3.n1.a5

Cited by 132 publications

(152 citation statements)

References 43 publications

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“…In addition, when the computational domain is a disk in 2D, and resp., a cylinder in 3D, the SIFD discretization can be extremely efficient in practical computation by using polar coordinates in 2D, and resp., cylindrical coordinates in 3D, together with fast direct Poisson solver. A similar idea to this method has been used in simulating quantized vortex dynamics in rotating BEC [6,9,12].…”

Section: Discussionmentioning

confidence: 99%

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Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation

2012

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Abstract. We analyze finite difference methods for the Gross-Pitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate, for the conservative Crank-Nicolson finite difference (CNFD) method and semi-implicit finite difference (SIFD) method, at the order of O(h 2 + τ 2 ) in the l 2 -norm and discrete H 1 -norm with time step τ and mesh size h. Besides the standard techniques of the energy method, the key technique in the analysis for the SIFD method is to use the mathematical induction, and resp., for the CNFD method is to obtain a priori bound of the numerical solution in the l ∞ -norm by using the inverse inequality and the l 2 -norm error estimate. In addition, for the SIFD method, we also derive error bounds on the errors between the mass and energy in the discretized level and their corresponding continuous counterparts, respectively, which are at the same order of the convergence rate as that of the numerical solution itself. Finally, numerical results are reported to confirm our error estimates of the numerical methods.

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Section: Discussionmentioning

confidence: 99%

“…In this paper, we analyze different finite difference approximations of the GrossPitaevskii equation (GPE) with an angular momentum rotation term in ddimensions (d = 2, 3) for modeling a rotating Bose-Einstein condensate (BEC) [35,12]:…”

Section: Introductionmentioning

confidence: 99%

Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation

2012

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“…Inserting (20) into (16) and collecting real and imaginary parts, we get the transport equation for ρ ε and the Hamilton-Jacobi equation for the phase S ε :…”

Section: Semiclassical Scaling and Geometrical Opticsmentioning

confidence: 99%

“…(21) is the transport equation for the atom density and (22) the Hamilton-Jacobi equation for the phase. Furthermore, by defining the current density [20,15] …”

Section: Semiclassical Scaling and Geometrical Opticsmentioning

confidence: 99%

“…S-shaped vortex and U-shaped vortex, numerically in cigar-shaped condensation and compared with experimental results [66], Garcia-Ripoll and Perez-Garcia [45,44,47], Bao and Zhang [21] studied stability of the central vortex, Tsubota et. al [75] reported vortex lattice formation, Bao et al [9,20] presented a continuous normalized gradient flow with backward Euler finite difference discretization to compute ground state, provided asymptotics of the energy and chemical potential of the ground state in the semiclassical regime and showed that the ground state is a global minimizer of the energy functional over the unit sphere and all excited states are saddle points in the linear case. Moreover, Svidzinsky and Fetter [72] have studied dynamics of a vortex line depending on its curvature.…”

Section: Introductionmentioning

confidence: 99%

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Ground states and dynamics of rotating Bose-Einstein condensates

Bao

Transport Phenomena and Kinetic Theory

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In this paper, we review the mathematical and numerical studies of ground states and dynamics in rotating Bose-Einstein condensates (BEC). We start from the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term, scale it to obtain a four-parameter model, reduce it to a 2D GPE in the limiting regime of strong anisotropic confinement and present its semiclassical scaling and geometrical optics. We discuss existence/nonexistence problem for ground states depending on the angular velocity. We examine the conservation of the angular momentum expectation and the condensate width and analyze the dynamics of a stationary state with a shift in its center. Finally, numerical methods for computing ground state and dynamics of rotating BEC are reviewed and some numerical results are reported.

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Supersymmetric Hamiltonian and Vortex Formation Model in a Quantum Nonlinear System in an Inhomogeneous Electromagnetic Field

Manzetti

Trounev

2019

Advcd Theory and Sims

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Vortices and quasiparticles in 2D systems are studied for the last decades in relation to the development of technologies in quantum electromagnetics, optics, and quantum computation. Hamiltonians are a fundamental part for studying quasiparticles and vortices, and provide models to calculate the eigenvalues and eigenfunctions that describe a real physical state. By devising supersymmetry, a Hamiltonian for the description of vortices is developed forming in an 2D electron gas and study the numerical solutions under the effects of an alternating electromagnetic field. The numerical analysis shows that vorticity is formed spontaneously without symmetry-breaking and vortices arise from the boundaries and converge toward the center of the system in a similar fashion to natural hydrodynamic phenomena in water or plasma. Additionally, the equation under damping conditions is studied, a homogenous magnetic field and under the absence of an electromagnetic field. The results and the study of the parameters indicate that the supersymmetic wave equation (SWE) may be a good model equation to describe vorticity for quantum electromagnetics, hydrodynamics, and other physical phenomena in the realm of physical and quantum physical sciences. levels of n. The model is similar to the Laughlin model which describes the formation of quasiparticles in a quantum hall, [2] however it acts as an extension in describing vorticity and quantum rotation. These vortices are in essence Bose-Einstein condensates [3] which arise at ultra-low temperatures within a gas such as Helium, also known as superfluids or quantum fluids. This phenomenon gives rise to properties such as superfluidity and superconductivity in gases and metals, where a common quantum state arises for holes with the same vorticity, and hence lead to a uniform band for superconductivity and superfluidity. This phenomenon is of particular appeal to quantum computing and quantum optics. [4][5][6][7] Additionally, quantum computing and optical technologies can be developed further also by constructing systems with alternating magnetic fields, which provide an auxiliary effect on the vortices and generates quantum wave dynamics that trigger exotic properties of appeal to the scientific community. For instance, systems of particles subjected to impulse (staggered) magnetic fields have been studied in ultra-cold atoms recently. [8] The study shows that electronic band structures formed under the effects of the variable magnetic field affect the shape of the Dirac cones formed in the conduction band, hence leading to changes in conductivity. Interestingly, the same study shows that the effect of a staggered magnetic field on bosons generates superfluid phases [8] leading to zero viscosity and therefore zero resistance in the optical band. Furthermore, a second study, [9] shows that staggered electromagnetic fields lead to the formation of uniform superfluids and Mott insulating phases [10] and a novel kind of finite-momentum superfluid phase, directly tunable by the quantized stagg...

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Ground, Symmetric and Central Vortex States in Rotating Bose-Einstein Condensates

Cited by 132 publications

References 43 publications

Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation

Optimal error estimates of finite difference methods for the Gross-Pitaevskii equation with angular momentum rotation

Ground states and dynamics of rotating Bose-Einstein condensates

Supersymmetric Hamiltonian and Vortex Formation Model in a Quantum Nonlinear System in an Inhomogeneous Electromagnetic Field

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