Pair Creation in Space–Time Modulated Media: Sector Rules and a Silent Zero Mode

A medium modulated in time creates photon pairs from the vacuum through the symplectic map of classical parametric gain. This article states which pairs grow without bound in non-uniform media. A space–time symmetry combining a spatial operation with a fraction of the modulation period gives every mode a sector label. A pair or conversion channel can open only between modes of equal label; otherwise the Floquet multipliers cross exactly, at every depth reached from the unmodulated medium while the continued labels stay simple. For a mirror with a half-period shift the two parities must multiply to (−1)^q at the q-th harmonic. Around a pump vortex of charge ℓ on a ring of N resonators, pair angular momenta add to qℓ modulo N. A staggered single-harmonic pump creates growing pairs only at odd harmonics, unless the spectrum is commensurate. When a single harmonic modulates the inertial term, such as a permittivity or a capacitance, periodic solutions have a silent zero-frequency component, and their space splits into two complex-conjugate halves. A degenerate channel at an even harmonic then has zero width unless the spectrum is commensurate. This closure and the sector rule forbid growing single-mode squeezing at every harmonic of a mirror-even cavity with a mirror-odd modulation and a non-commensurate spectrum. The silent zero mode alone closes the even momentum gaps of photonic time crystals with sinusoidal permittivity. A second harmonic or a modulated restoring term reopens the channels closed by the silent zero mode, and a symmetry defect those closed by the sector rule, at most linearly. A commensurate mode of the same sector reopens the degenerate channel, at a rate measured to grow as d^4 with the depth d. The results are proved for linear, lossless media and measured on cavity, ring and random models, not on a fabricated device.