A theory of dissipative nonlinear conductivity, $sigma_1(omega,H)$, of s-wave superconductors under strong electromagnetic fields at low temperatures is proposed. Closed-form expressions for $sigma_1(H)$ and the surface resistance $R_s(omega,H)$ are obtained in the nonequilibrium dirty limit for which $sigma_1(H)$ has a significant minimum as a function of a low-frequency $(hbaromegall k_BT)$ magnetic field $H$. The calculated microwave suppression of $R_s(H)$ is in good agreement with recent experiments on alloyed Nb resonator cavities. It is shown that superimposed dc and ac fields, $H=H_0+H_acosomega t$, can be used to reduce ac dissipation in thin film nanostructures by tuning $sigma_1(H_0)$ with the dc field.
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