Solitons are among the most distinguishing fundamental excitations in a wide range of non-linear systems such as water in narrow channels, high speed optical communication, molecular biology and astrophysics. Stabilized by a balance between spreading and focusing, solitons are wavepackets, which share some exceptional generic features like form-stability and particle-like properties. Ultracold quantum gases represent very pure and well-controlled non-linear systems, therefore offering unique possibilities to study soliton dynamics. Here we report on the first observation of long-lived dark and dark-bright solitons with lifetimes of up to several seconds as well as their dynamics in highly stable optically trapped 87 Rb Bose-Einstein condensates. In particular, our detailed studies of dark and dark-bright soliton oscillations reveal the particle-like nature of these collective excitations for the first time. In addition, we discuss the collision between these two types of solitary excitations in Bose-Einstein condensates. The dynamics of non-linear systems plays an essential role in nature, ranging from strong non-linear interactions of elementary particles to non-linear wave phenomena in oceanography and meteorology. A special class of non-linear phenomena are solitons with interesting particle and wave-like behaviour. Reaching back to the observation of "waves of translation" in a narrow water channel by Scott-Russell in 1834 [1], solitons are nowadays recognized to appear in various systems as different as astrophysics, molecular biology and non-linear optics [2]. They are characterized as localized solitary wavepackets that maintain their shape and amplitude caused by a self-stabilization against dispersion via a non-linear interaction. While an early theoretical explanation of this non-dispersive wave phenomenon was given by Korteweg and de Vries in the late 19th century it was not before 1965 that numerical simulations of Zabusky and Kruskal theoretically proved that these solitary waves preserve their identity in collisions [3,4]. This revelation led to the term "soliton" for this type of collective excitation. Nowadays solitons are a very active field of research in many areas of science. In the field of non-linear optics they attract enormous attention due to applications in fast data transfer. Bose-Einstein condensates (BEC) of weakly interacting atoms offer fascinating possibilities for the study of nonlinear phenomena, as they are very pure samples of ultracold gases building up an effective macroscopic wave function of up to mm size. Non-linear effects like collective excitations [5], four-wave mixing [6] and vortices [7,8] have been studied, to name only a few examples. The existence and some fundamental properties of solitons have been deduced from few experiments employing ultra-cold quantum gases. Bright solitons, characterized as non-spreading matter-wave packets, have been observed in BEC with attractive interaction [9,10,11] where they represent the ground state of the system. In a repulsively...

We present experimental data showing the head-on collision of dark solitons generated in an elongated Bose-Einstein condensate. No discernable interaction can be recorded, in full agreement with the fundamental theoretical concepts of solitons as mutually transparent quasiparticles. Our soliton generation technique allows for the creation of solitons with different depths; hence, they can be distinguished and their trajectories be followed. Simulations of the 1D-Gross-Pitaevskii equation have been performed to compare the experiment with a mean-field description.

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Two-component mixtures in optical lattices reveal a rich variety of different phases. We employ an exact diagonalization method to obtain the relevant correlation functions in hexagonal optical lattices to characterize those phases. We relate the occupation difference of the two species to the magnetic polarization. 'Iso'-magnetic correlations disclose the nature of the system, which can be of easy-axis type, bearing phase segregation, or of easy-plane type, corresponding to super-counter-fluidity. In the latter case, the correlations reveal easy-plane segregation, involving a highly-entangled state. We identify striking correlated supersolid phases appearing within the superfluid limit.