Symmetry breaking and spatiotemporal pattern formation in photonic time crystals

In this work, we explore the dynamics of time-varying photonic media with an optical Kerr nonlinearity and an associated phase transition. The interplay between a periodically modulated permittivity and the nonlinearity induces a continuous transition of electromagnetic waves to a state with broken spatial and time translation symmetries. This transition gives rise to a lattice-like wave pattern, in many ways similar to a spatial crystallization in solids. Symmetry breaking triggers the emergence of soft, Goldstone-like modes, which propagate as deformations of the lattice structure, as well as massive Higgs-like modes—spatially uniform oscillations of the field amplitude. We extend the analysis of the nonequilibrium symmetry breaking to <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mn>2</a:mn><a:mo>+</a:mo><a:mn>1</a:mn></a:math>-dimensional time-varying media and discuss pattern formation as well as the connection to discrete dissipative time crystals.