Classical Petermann Factor as a Measure of Quantum Squeezing in Photonic Time Crystals

Photonic time crystals realize a continuum of momentum-resolved SU(1,1) parametric amplifiers. We show that a classical quantity, the Petermann factor of the effective Floquet Bogoliubov-de Gennes (BdG) dynamical matrix, sets the scale of their quantum noise. In stable bands it fixes the Bogoliubov mixing and the vacuum quasiparticle population, while in momentum gaps it sets the photon-number prefactor and enhances the squeezing dynamics, with the Floquet growth rate setting the time scale. This converts classical measurements of mode nonorthogonality into quantitative predictions for squeezing and photon generation, and offers a compact design parameter for engineering quantum resources in two-mode BdG platforms.

💡 Research Summary

The manuscript establishes a direct quantitative bridge between a classical non‑Hermitian metric—the Petermann factor (PF)—and quantum squeezing in photonic time crystals (PTCs). PTCs are media whose permittivity ε(t) is periodically modulated in time, thereby realizing a continuum of momentum‑resolved SU(1,1) parametric amplifiers. The authors start from a quantum Hamiltonian for a pair of opposite‑momentum modes (k, −k):

H_k(t)=A(t)(n_k+n_{−k}+1)+B(t)(a_k a_{−k}+a_k† a_{−k}†),

with A(t) and B(t) determined by the instantaneous inverse permittivity. By introducing the Nambu spinor Φ=(a_k, a_{−k}†)^T, the Hamiltonian assumes a Bogoliubov‑de Gennes (BdG) form H_BdG(t)=(\begin{pmatrix}A(t)&B(t)\B(t)&A(t)\end{pmatrix}). The Heisenberg equation yields the dynamical matrix M_q(t)=σ_z H_BdG(t), which is periodic with the modulation period T=2π/Ω.

Because the classical wave equation for the same time‑modulated medium shares the identical monodromy matrix U(T,0)=𝒯 exp