Quantum derivatives and high-frequency gain in semiconductor   superlattices
  We derive simple difference formulas describing a small-signal absorption of THz field in a semiconductor superlattice driven by a microwave pump. We give a transparent geometric interpretation of these formulas that allows a search of optimum conditions for the gain employing only a simple qualitative analysis. Our theoretical approach provides a powerful tool for finding the correspondence between quasistatic and dynamic regimes in ac-driven semiconductor superlattices. 
A. V. Shorokhov and K. N. Alekseev