Aspects of quantum geometry in photonic 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 Möbius isometry on the Poincaré 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.

Figures (5)

• Figure 1: (a) Band structure of the system under a temporally periodic modulation of the permittivity defined in Eq. (\ref{['epsilon']}). The black dots denote the real part of the quasifrequency, while the gray dots represent its imaginary part. The blue diamonds, labeled with Roman numerals I–V, mark the points $k = \{0.531, 0.540, 0.731, 0.887, 1.169\}$, listed in the same order as the labels. (b) Temporal profile of the step-like modulation over one period $T$, the permittivity takes the value $\varepsilon_1 = 3$ during the first half of the cycle ($0.5T$) and $\varepsilon_2 = 1$ during the second half. This periodic driving generates the band structure shown in the left panel.
• Figure 2: Top panels: Coherent state dynamics corresponding to points I-V (see FIG.\ref{['fig:1']}) in the form of stroboscopic (with the period $T$) trajectories of the projective parameter $z(t)$ on the Poincaré disk. The points $\gamma_\pm$ correspond to fixed points of the Möbius map $\mathcal{M}_k$. Proximity to the boundary of the disk corresponds to the number of entangled pairs. At exceptional points (e.g. I), coherent states approach the fixed points $\gamma_+=\gamma_-$ in a critically slow way. States in the gap (e.g. II) follow hyperbolic trajectories approaching one of two fixed points on the boundary. Elliptic trajectories (e.g. III-V) instead encircle a unique fixed point (for $|\,z\,|<1$). Upon approaching a critical point (III), the states begin to exhibit a critical slowing down. For incommensurate $\Omega_k$ and $\omega=2\pi/T$ (IV), trajectories display quasi-periodic behaviour. In the bottom panels, the entanglement entropy is plotted at stroboscopic times.
• Figure 3: (a) Phase diagram for the photonic time crystal: entanglement entropy $S_k$ can be used as an indicator of instability. (b),(c) For $k=1.22$, the drive parameter $T_1/T$ is varied across an exceptional point. Color bar corresponds to the trace of the monodromy matrix, ${\rm Tr}M_k$. The quasi-energy curve is non-analytic at exceptional points (at ratios $T_1/T$ indicated by red lines), indicating a phase transition. In all panels the bipartite entropy after $n=40$ periods, starting with the vacuum coherent state, is plotted as a function of $T_1/T$.
• Figure 4: (a) Geometric (Aharonov-Anandan) phase $\Gamma_k$ accumulated in the course of cyclic evolution about the stroboscopic fixed points plotted in (b). (c) Hyperbolic radii of stroboscopic trajectories starting from the vacuum state. When $R_k=0$ (at $k$ indicated by a dashed line), the vacuum is a Floquet state, representing a transparency to time refraction in the stroboscopic sense.
• Figure S1: Stroboscopic features of coherent state evolution starting from the quasi-vaccuum state for $\Omega_k=\omega/2n$, (a) $n=3$, (b) $n=4$