  Spontaneous symmetry breaking (SSB) is key for our understanding of phase
transitions and the spontaneous emergence of order. Photonics provide versatile
systems to study SSB. In this work, we report that for a two-dimensional (2D)
periodic nonlocal metasurface with gain, SSB occurs in the lasing transition,
breaking parity symmetry. We study diffractive hexagonal plasmon nanoparticle
lattices, where the K-points in momentum space provide two modes that are
exactly degenerate in frequency and identically distributed in space. Using
femtosecond pulses to energize the gain medium, we simultaneously capture
single shot realspace and wavevector resolved Fourier images of laser emission.
By combining Fourier- and real-space, we resolve the two order parameters for
which symmetry breaking simultaneously occurs: spatial parity and U(1)
(rotational) symmetry breaking, evident respectively as random relative mode
amplitude and phase. Thereby, we quantify for the first time SSB in 2D periodic
metasurfaces. These currently receive much interest as experimentally
accessible implementations of seminal solid-state physics Hamiltonians and
provide a large design space for exploring SSB in scenarios with different
symmetries, mode degeneracies and topological properties. The methodology
reported in this work is generally applicable to 2D plasmonic and dielectric
metasurfaces and opens numerous opportunities for the study of SSB and
emergence of spatial coherence in metaphotonics.

Optics