The successful emulation of the Hubbard model [1] in optical lattices [2,3] has stimulated world wide efforts to extend their scope to also capture more complex, incompletely understood scenarios of many-body physics [4,5]. Unfortunately, for bosons, Feynmans fundamental "nonode" theorem [6] under very general circumstances predicts a positive definite ground state wave function with limited relevance for manybody systems of interest.A promising way around Feynmans statement is to consider higher bands in optical lattices with more than one dimension [7], where the orbital degree of freedom with its intrinsic anisotropy due to multiple orbital orientations gives rise to a structural diversity, highly relevant, for example, in the area of strongly correlated electronic matter. In homogeneous two-dimensional optical lattices, lifetimes of excited bands on the order of a hundred milliseconds are possible [8,9] but the tunneling dynamics appears not to support cross-dimensional coherence [8,10]. Here we report the first observation of a superfluid in the P -band of a bipartite optical square lattice with S-orbits and P -orbits arranged in a chequerboard pattern. This permits us to establish full cross-dimensional coherence with a life-time of several ten milliseconds. Depending on a small adjustable anisotropy of the lattice, we can realize real-valued striped superfluid order parameters with different orientations P x ± P y or a complex-valued P x ± iP y order parameter, which breaks time reversal symmetry and resembles the π-flux model proposed in the context of high temperature superconductors [11]. Our experiment opens up the realms of orbital superfluids to investigations with optical lattice models.Orbital optical lattices permit to prepare exciting new many-body scenarios with relevant counterparts in the area of strongly correlated electronic matter, where the orbital degree of freedom with its rich structure of possible orientations gives rise to a wealth of interesting phenomena. For example, in the transition-metal oxides orbital physics is believed to be a key element for understanding their metal-insulator transitions, superconductivity, or colossal magnetoresistance. Recent theoretical proposals have considered, for example, the formation of multiflavor and multiorbital systems [8,9,12,13], supersolid quantum phases in cubic lattices [14,15], quantum stripe ordering in triangular lattices [16], or Wigner crystallization in honeycomb lattices [17]. The recent prediction, that the life time of atoms in higher Bloch bands can be unexpectedly long, has further strengthened the potential significance of orbital physics in optical lattices [8,9]. Previous experiments have demonstrated that the excitation of higher Bloch bands is in fact possible [18,19]. In a recent experiment, investigating a homogeneous quasi one-dimensional (1D) lattice of bosons, transient partial coherence in the P -band has been observed [10]. In an extension to a two-dimensional square lattice, however, no cross-coherence could be est...

We report on the first observation of bosons condensed into the energy minima of an F -band of a bipartite square optical lattice. Momentum spectra indicate that a truly complex-valued staggered angular momentum superfluid order is established. The corresponding wave function is composed of alternating local F 2x 3 −3x + i F 2y 3 −3y -orbits and local S-orbits residing in the deep and shallow wells of the lattice, which are arranged as the black and white areas of a checkerboard. A pattern of staggered vortical currents arises, which breaks time reversal symmetry and the translational symmetry of the lattice potential. We have measured the populations of higher order Bragg peaks in the momentum spectra for varying relative depths of the shallow and deep lattice wells and find remarkable agreement with band calculations.

The study of superconductivity with unconventional order is complicated in condensed matter systems by their extensive complexity. Optical lattices with their exceptional precision and control allow one to emulate superfluidity avoiding many of the complications of condensed matter. A promising approach to realize unconventional superfluid order is to employ orbital degrees of freedom in higher Bloch bands. In recent work, indications were found that bosons condensed in the second band of an optical chequerboard lattice might exhibit p x ± i p y order. Here we present experiments, which provide strong evidence for the emergence of p x ± i p y order driven by the interaction in the local p-orbitals. We compare our observations with a multi-band Hubbard model and find excellent quantitative agreement.

We report on the condensation of bosons in the 4th band of an optical checkerboard lattice providing a topologically induced avoided band crossing involving the 2nd, 3rd, and 4th Bloch bands. When the condensate is slowly tuned through the avoided crossing, accelerated band relaxation arises and the zero momentum approximately C4-invariant condensate wave function acquires finite momentum order and reduced C2 symmetry. For faster tuning Landau-Zener oscillations between different superfluid orders arise, which are used to characterize the avoided crossing.

Approximate solutions of the Gross-Pitaevskii (GP) equation, obtained upon neglection of the kinetic energy, are well known as Thomas-Fermi solutions. They are characterized by the compensation of the local potential by the collisional energy. In this article we consider exact solutions of the GP-equation with this property and definite values of the kinetic energy, which suggests the term "kinetic Thomas-Fermi" (KTF) solutions. We point out that a large class of light-shift potentials gives rise to KTF-solutions. As elementary examples, we consider one-dimensional and two-dimensional optical lattice scenarios, obtained by means of the superposition of two, three and four laser beams, and discuss the stability properties of the corresponding KTF-solutions. A general method is proposed to excite two-dimensional KTF-solutions in experiments by means of time-modulated light-shift potentials.