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

Abstract

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.

Figures (1)

• Figure 1: (a) Petermann factor of a PTC with $\varepsilon^{-1}(t)=1-2\alpha\cos{\Omega t}$, where $\alpha=0.15$ and $\Omega=1$. The blue shaded region indicates the MG (unstable) regime, and $K_k$ diverges at the MG edges $k_{c\pm}$. The dashed line represents the lower bound $K_k=1$. (b) Squeezing parameter $r_k$ and the quasiparticle expectation value in the photon vacuum, $\braket{\hat{n}_{k,b}}$ in the band (stable) regime. As the momentum approaches the MG edge $k_{c-}$, both quantities increase with the Petermann factor $K_k$, as quantified by Eqs. (\ref{['eq8']}) and (\ref{['eq9']}). (c) Temporal evolution of the photon number $\braket{\hat{n}_k(t)}$ in the MG, obtained from the effective Floquet Hamiltonian. The curves represent stroboscopic evolution at integer multiples of $T$. Legends indicate the momentum ratio $k/k_{c-}$. As the system approaches the critical point ($k\rightarrow k_{c-}$), the dynamics follow the universal quadratic scaling $|\Delta_{k_{c-}}|^2t^2$, illustrated by the red dashed line.