  The successful emulation of the Hubbard model in optical lattices has
stimulated world wide efforts to extend their scope to also capture more
complex, incompletely understood scenarios of many-body physics. Unfortunately,
for bosons, Feynmans fundamental "no-node" theorem under very general
circumstances predicts a positive definite ground state wave function with
limited relevance for many-body systems of interest. A promising way around
Feynmans statement is to consider higher bands in optical lattices with more
than one dimension, 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 but the
tunneling dynamics appears not to support cross-dimensional coherence. 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 $P_x \pm P_y$ or a
complex-valued $P_x \pm i P_y$ order parameter, which breaks time reversal
symmetry and resembles the $\pi$-flux model proposed in the context of high
temperature superconductors. Our experiment opens up the realms of orbital
superfluids to investigations with optical lattice models.
