  We introduce the notion of steady-state quantum chaos as a general phenomenon
in open quantum many-body systems. Classifying an isolated or open quantum
system as integrable or chaotic relies in general on the properties of the
equations governing its time evolution. This however may fail in predicting the
actual nature of the quantum dynamics, that can be either regular or chaotic
depending on the initial state. Chaos and integrability in the steady state of
an open quantum system are instead uniquely determined by the spectral
structure of the time evolution generator. To characterize steady-state quantum
chaos we introduce a spectral analysis based on the spectral statistics of
quantum trajectories (SSQT). We test the generality and reliability of the SSQT
criterion on several dissipative systems, further showing that an open system
with chaotic structure can evolve towards either a chaotic or integrable steady
state. We study steady-state chaos in the driven-dissipative Bose-Hubbard
model, a paradigmatic example of out-of-equilibrium bosonic system without
particle number conservation. This system is widely employed as a building
block in state-of-the-art noisy intermediate-scale quantum devices, with
applications in quantum computation and sensing. Finally, our analysis shows
the existence of an emergent dissipative quantum chaos, where the classical and
semi-classical limits display an integrable behaviour. This emergent
dissipative quantum chaos arises from the quantum and classical fluctuations
associated with the dissipation mechanism. Our work establishes a fundamental
understanding of the integrable and chaotic dynamics of open quantum systems
and paves the way for the investigation of dissipative quantum chaos and its
consequences on quantum technologies.
