  We derive a Born-Markov master equation describing the dissipation induced by
a bath of lossy but coherently driven two-level systems (TLS) coupled to a
bosonic system via Jaynes-Cummings interaction. We analytically derive all the
master equation rates. We characterize these rates for the particular case of a
single-mode system coupled to identical TLS. We study the steady state of the
system and its exotic properties stemming from the non-thermal stationary state
of the driven TLS bath. These properties include dissipative amplification,
bath-induced linear instability, and both coherent and dissipative squeezing.
The master equation is valid for arbitrarily strong TLS driving, and it can be
generalized to include multi-level systems or other system-bath interaction
terms, among others. Our work provides a tool to study and characterize
TLS-induced decoherence, a key limiting factor in quantum technological devices
based on, for instance, superconducting circuits, magnonic systems, or quantum
acoustics.
