Abstract. In this paper we improve traditional steepest descent methods for the direct minimization of the Gross-Pitaevskii (GP) energy with rotation at two levels. We first define a new inner product to equip the Sobolev space H 1 and derive the corresponding gradient. Secondly, for the treatment of the mass conservation constraint, we use a new projection method that avoids more complicated approaches based on modified energy functionals or traditional normalization methods. The descent method with these two new ingredients is studied theoretically in a Hilbert space setting and we give a proof of the global existence and convergence in the asymptotic limit to a minimizer of the GP energy. The new method is implemented in both finite difference and finite element twodimensional settings and used to compute various complex configurations with vortices of rotating Bose-Einstein condensates. The new Sobolev gradient method shows better numerical performances compared to classical L 2 or H 1 gradient methods, especially when high rotation rates are considered.

Quantum carpets-the spatio-temporal de Broglie density profiles-woven by an atom or an electron in the near-field region of a diffraction grating bring to light, in real time, the decoherence of each individual component of the interference term of the Wigner function characteristic of superposition states. The proposed experiments are feasible with present-day technology.

In this contribution, we prove the global existence and uniqueness of a trajectory that globally converges to the minimizer of the Gross-Pitaevskii energy functional for a large class of external potentials. Using the method of Sobolev gradients, we can provide an explicit construction of this minimizing sequence.Based upon this theoretical framework we numerically apply these results to a specific realization of the external potential and illustrate the main benefits of the method of Sobolev gradients, which are high numerical stability and rapid convergence toward the minimizer.

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