  The rich physical properties of multiatomic molecules and crystalline
structures are determined, to a significant extent, by the underlying geometry
and connectivity of atomic orbitals. This orbital degree of freedom has also
been used effectively to introduce structural diversity in a few synthetic
materials including polariton lattices nonlinear photonic lattices and
ultracold atoms in optical lattices. In particular, the mixing of orbitals with
distinct parity representations, such as $s$ and $p$ orbitals, has been shown
to be especially useful for generating systems that require alternating phase
patterns, as with the sign of couplings within a lattice. Here we show that by
further breaking the symmetries of such mixed-orbital lattices, it is possible
to generate synthetic magnetic flux threading the lattice. This capability
allows the generation of multipole higher-order topological phases in synthetic
bosonic platforms, in which $\pi$ flux threading each plaquette of the lattice
is required, and which to date have only been implemented using tailored
connectivity patterns. We use this insight to experimentally demonstrate a
quadrupole photonic topological insulator in a two-dimensional lattice of
waveguides that leverage modes with both $s$ and $p$ orbital-type
representations. We confirm the nontrivial quadrupole topology of the system by
observing the presence of protected zero-dimensional states, which are
spatially confined to the corners, and by confirming that these states sit at
the band gap. Our approach is also applicable to a broader range of
time-reversal-invariant synthetic materials that do not allow for tailored
connectivity, e.g. with nanoscale geometries, and in which synthetic fluxes are
essential.
