Aspects of quantum geometry in photonic time crystals

TL;DR

This paper introduces a geometric framework for understanding quantum light in photonic time crystals, linking stability, topology, and phase through hyperbolic geometry on the Poincaré disk.

Contribution

It develops a novel geometric description of quantum photonic modes using SU(1,1) coherent states and relates topological features to stability regimes in photonic time crystals.

Findings

Trajectories classified as stable, unstable, or critical based on topology.

Geometric phase connected to hyperbolic area in the projective Hilbert space.

Provides an intuitive geometric view of quantum stability in time crystals.

Abstract

We develop a geometric description of quantum light in photonic time crystals on the SU(1,1) coherent-state manifold. In a projective picture, the evolution of each mode appears as a M\"obius isometry on the Poincar\'e disk, where topologies of trajectories distinguish stable, unstable, and critical regimes. The geometric phase is related to the hyperbolic area enclosed by cyclic paths in the complex projective Hilbert space. This framework offers an intuitive view of stability and topology in quantum photonic time crystals.