Pair Creation in Space–Time Modulated Media II: The Commensurate Rate and Twin Channels

In a space–time symmetric medium whose inertial term is modulated by a single harmonic, the even degenerate pair channels stay closed as long as the spectrum is not commensurate with the drive. When a second mode sits at twice the frequency of the first and they convert at the first harmonic, the closed channel of the first reopens at fourth order in the modulation depth d. This article gives the rate. The four resonant components of the Fourier chain form a Hermitian 4×4 nonlinear pencil. Its two halves couple only through the silent zero mode. The largest growth rate is c_4d^4, where c_4 is a closed expression in the modal couplings. The correction is of order d^2. The Hill form and the one-period map confirm the coefficient in 2 cavities. Under detuning the d^4 law turns into a d^3 law, in proportion to the dressed detuning, as the tongue becomes the pair of the 2 modes at the third harmonic. A closed form gives the rate, the centre of the tongue and its width in drive frequency, which is 4/3 of the rate. The second part concerns twin channels. A time-reversing symmetry followed by complex conjugation maps every open channel onto an open channel with the same Floquet multiplier. The image is the channel itself, its conjugate, or a twin. A conjugate twin, such as the intraband pair (k,−k) of a uniform medium, has a real multiplier. A twin doubles the multiplier quadruplet. On a chiral ring with a time-symmetric pump the umklapp pairs in sectors with a complex label come in twins, although no mirror relates them. A defect that breaks the time reversal splits twin rates linearly. A glide time reversal whose square is −1 doubles every channel of its sector. Design rules and refutable predictions follow.