  We numerically study the spectral statistics of open quantum many-body
systems (OQMBS) as signatures of quantum chaos (or the lack thereof), using the
dissipative spectral form factor (DSFF), a generalization of the spectral form
factor to complex spectra. We show that the DSFF of chaotic OQMBS generically
displays the $\textit{quadratic}$ ramp-plateau behaviour of the Ginibre
ensemble from random matrix theory, in contrast to the linear ramp-plateau
behaviour of the Gaussian ensemble in closed quantum systems. Furthermore, in
the presence of many-body interactions, such RMT behaviour emerges only after a
time scale $\tau_{\mathrm{dev}}$, which generally increases with system size
for sufficiently large system size, and can be identified as the non-Hermitian
analogue of the $\textit{many-body Thouless time}$. The universality of the
random matrix theory behavior is demonstrated by surveying twelve models of
OQMBS, including random Kraus circuits (quantum channels) and random
Lindbladians (Liouvillians) in several symmetry classes, as well as
Lindbladians of paradigmatic models such as the Sachdev-Ye-Kitaev (SYK), XXZ,
and the transverse field Ising models. We devise an unfolding and filtering
procedure to remove variations of the averaged density of states which would
otherwise hide the universal RMT-like signatures in the DSFF for chaotic OQMBS.
Beyond chaotic OQMBS, we study the spectral statistics of non-chaotic OQMBS,
specifically the integrable XX model and a system in the many-body localized
(MBL) regime in the presence of dissipation, which exhibit DSFF behaviours
distinct from the ramp-plateau behaviour of random matrix theory. Lastly, we
study the DSFF of Lindbladians with the Hamiltonian term set to zero, i.e. only
the jump operators are present, and demonstrate that the results of RMT
universality and scaling of many-body Thouless time survive even without
coherent evolution.
