I report on a recent joint work together with R. Carles and P. Markowich, where we consider the time-dependent Gross-Pitaevskii equation modeling Bose-Einstein condensation of trapped dipolar quantum gases. Existence and uniqueness as well as the possible blow-up of solutions are studied. Moreover, we discuss the problem of dimension-reduction for this nonlinear and nonlocal Schrödinger equation.

• Event: Workshop "The Gross-Pitaevskii equation and its application for BEC in optical lattices" (2008)

In this talk, we present efficient and stable numerical methods to compute ground states and dynamics of Bose-Einstein condensates (BEC) in a rotational frame. As preparatory steps, we take the 3D Gross-Pitaevskii equation (GPE) with an angular momentum rotation, scale it to obtain a four-parameter model and show how to reduce it to 2D GPE in certain limiting regimes. Then we study numerically and asymptotically the ground states, excited states and quantized vortex states as well as their energy and chemical potential diagram in rotating BEC. Some very interesting numerical results are observed. Finally, we study numerically stability and interaction of quantized vortices in rotating BEC. Some interesting interaction patterns will be reported. This talk is based on joint work with Qiang Du, Peter Markowich, Hanquan Wang and Yanzhi Zhang.

• Event: Workshop "The Gross-Pitaevskii equation and its application for BEC in optical lattices" (2008)

We investigate the dynamics of quasi-one-dimensional Bose-Einstein condensates (BECs) with spatially-periodic scattering lengths. This type of "collisionally inhomogeneous" BEC is described by a Gross-Pitaevskii (GP) equation with a nonlinearity coefficient that varies periodically in space. For the case of a sinusoidal coefficient, we examine the dynamics of spatially-extended states (modulated amplitude waves) that we construct analytically using the method of averaging. For the case of piecewise-constant coefficient, we examine the dynamics of solitary waves that we construct using a "stitching" technique.

• Event: Workshop "The Gross-Pitaevskii equation and its application for BEC in optical lattices" (2008)

In this talk, we introduce the Bloch-decomposition based time-splitting spectral method to conduct numerical simulations of the dynamics of (non)linear SchrÄodinger equations subject to periodic and confining potentials. We consider this system as a two-scale asymptotic problem with different scalings of the nonlinearity. In particular we discuss (nonlinear) mass transfer between different Bloch bands and also present three-dimensional simulations for lattice Bose-Einstein condensates in the superfluid regime. Joint work with Shi Jin, Peter A. Markowich, and Christof Sparber

• Event: Workshop "The Gross-Pitaevskii equation and its application for BEC in optical lattices" (2008)

I review existence of stationary and moving gap solitons in the Gross--Pitaevskii equation with a small periodic potential. These solitons are approximated by the explicit solutions of the coupled-mode system. We show, however, that exponentially decaying traveling solutions of the Gross--Pitaevskii equation do not generally exist in the presence of a periodic potential due to bounded oscillatory tails ahead and behind the moving solitary waves. The oscillatory tails are not accounted in the coupled-mode formalism and are estimated by using techniques of spatial dynamics and local center-stable manifold reductions. Existence of bounded traveling solutions of the Gross--Pitaevskii equation with a single bump surrounded by oscillatory tails on a large interval of the spatial scale is proven by using these techniques.

• Event: Workshop "The Gross-Pitaevskii equation and its application for BEC in optical lattices" (2008)

Using the continuous and the discrete time-dependent GPE, we probe the Bogoliubov spectrum of a Bose-Einstein condensate in a 1D optical lattice. The presence of the optical lattice not only reduces the sound velocity, but is also responsible for phenomena which arise due to the interplay of interactions and periodic potential, like dynamical instabilities for a moving condensate. Moreover, the time-dependent GPE allows to go beyond the Bogoliubov description by entering the non-linear regime, where one can observe saturation and self-trapping effects. Finally, we include an external trapping potential to simulate situations closer to the experimental ones.

• Event: Workshop "The Gross-Pitaevskii equation and its application for BEC in optical lattices" (2008)

The phenomenon of matter wave interference lies at the heart of quantum physics. It has been observed in many contexts in the limit of non-interacting particles as a single particle effect. We observe and control many-body matter wave interference, which is driven by nonlinear particle interactions. In a multipath matter wave interferometer the macroscopic many-body wave function of an interacting atomic Bose-Einstein condensate develops a regular interference pattern, allowing us to directly visualize the effect of interaction induced phase shifts as time progresses. We demonstrate coherence for the nonlinear phase evolution in a matter wave spin-echo-type experiment when we stop and reverse the interaction driven evolution by first nulling the strength of the interaction and then driving the multipath phase shifts with opposite signs by means of an external potential. Alternatively, we balance the effect of interactions by means of the external potential and observe persistent Bloch oscillations. If time permits, we discuss experiments in which we confine the matter wave sample to a one-dimensional geometry and study Bloch osciallations and transport in a one-dimensional gas.

• Event: Workshop "The Gross-Pitaevskii equation and its application for BEC in optical lattices" (2008)