Temporal PT symmetry alone is not enough for a real spectrum; in sinusoidally modulated media the PT-symmetric phase appears only when the imaginary modulation depth exceeds the real one, near gamma = 1.
Yablonovitch, Physical review letters 58, 2059 (1987)
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Parity-time symmetry phase transition in photonic time-modulated media
Temporal PT symmetry alone is not enough for a real spectrum; in sinusoidally modulated media the PT-symmetric phase appears only when the imaginary modulation depth exceeds the real one, near gamma = 1.