  The microwave absorption and noise caused by quantum two-level systems (TLS)
dramatically suppress the coherence in Josephson junction qubits that are
promising candidates for a quantum information applications. Microwave
absorption by TLSs is not clearly understood yet because of the complexity of
their interactions leading to the spectral diffusion. Here, the theory of the
non-linear absorption in the presence of spectral diffusion is developed using
the generalized master equation formalism. The theory predicts that the linear
absorption regime holds while a TLS Rabi frequency is smaller than their phase
decoherence rate. At higher external fields, a novel non-linear absorption
regime is found with the loss tangent inversely proportional to the intensity
of the field. The theory can be generalized to acoustic absorption and lower
dimensions realized in superconducting qubits.
