Ground states and dynamics of rotating Bose-Einstein condensates

Abstract

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.

The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

“…The radial symmetry was used heavily by the authors of [10] in order to establish (30), using a maximum principle due to Berestycki, Nirenberg, and Varadhan [35] (see also the discussion following (79) herein) together with an intersection-comparison type of argument, mostly taking advantage of (7). The importance of the positivity of η ε and the radial symmetry of W in deriving (30) can be naively seen from the following consideration.…”

“…The radial symmetry was used heavily by the authors of [10] in order to establish (30), using a maximum principle due to Berestycki, Nirenberg, and Varadhan [35] (see also the discussion following (79) herein) together with an intersection-comparison type of argument, mostly taking advantage of (7). The importance of the positivity of η ε and the radial symmetry of W in deriving (30) can be naively seen from the following consideration. If W (r ) 0 for all r > 0, using that η ε > 0 and the method of moving planes [108], we can infer that η ε (r ) 0 for all r > 0 (see also Proposition 2.1 in [152] for an approach via a radially-symmetric rearrangement argument which takes advantage of the minimizing character of η ε , or Lemma 2 in [49] which uses the stability of η ε ).…”

“…(Actually, estimate (30) was needed in a region of the form (R − δ, R + Cε 3 ) for some constants δ, C > 0.) Making use of the previously mentioned estimates on f ε near the boundary of D 0 , and of some new estimates away from D 0 , the authors of [10] proved that if the angular velocity Ω is below a critical speed…”

Universally‐Convergent Squared‐Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations

“…If one views the eigenvalues as propagation constants, and requires the eigenfunctions to have a fixed power(norm), then this linear eigenvalue problem becomes the same as a solitary-wave problem with a pre-specified power. To treat this type of problems, one can use the imaginary-time evolution method [9,10,11,12,13], where the solution is normalized at every step to keep the pre-specified power. However, the problem with the imaginary-time evolution method is that it often diverges when the solution crosses zero [13].…”

“…This method can become cumbersome as well in high dimensions when the linear eigenvalue problem becomes harder to solve. Two more methods are the Petviashvili method [5,6,7,8] and the imaginary-time evolution method [9,10,11,12,13]. The convergence properties of the former method were studied in [6] for a class of equations with power nonlinearity, while those of the latter method were obtained in [13] for a much larger class of equations with arbitrary nonlinearity.…”