  The topological phases of matter are characterized using the Berry phase, a
geometrical phase, associated with the energy-momentum band structure. The
quantization of the Berry phase, and the associated wavefunction polarization,
manifest themselves as remarkably robust physical observables, such as
quantized Hall conductivity and disorder-insensitive photonic transport.
Recently, a novel class of topological phases, called higher-order topological
phases, were proposed by generalizing the fundamental relationship between the
Berry phase and the quantized polarization, from dipole to multipole moments.
Here, we demonstrate the first photonic realization of the quantized quadrupole
topological phase, using silicon photonics. In this 2nd-order topological
phase, the quantization of the bulk quadrupole moment in a two-dimensional
system manifests as topologically robust corner states. We unambiguously show
the presence of localized corner states and establish their robustness against
certain defects. Furthermore, we contrast these topological states against
topologically-trivial corner states, in a system without bulk quadrupole
moment, and observe no robustness. Our photonic platform could enable the
development of robust on-chip classical and quantum optical devices with
higher-order topological protection.
