  We propose the Sachdev-Ye-Kitaev Lindbladian as a paradigmatic solvable model
of dissipative many-body quantum chaos. It describes $N$ strongly coupled
Majorana fermions with random all-to-all interactions, with unitary evolution
given by a quartic Hamiltonian and the coupling to the environment described by
$M$ quadratic jump operators, rendering the full Lindbladian quartic in the
Majorana operators. Analytical progress is possible by developing a dynamical
mean-field theory for the Liouvillian time evolution on the Keldysh contour. By
disorder-averaging the interactions, we derive an (exact) effective action for
two collective fields (Green's function and self-energy). In the large-$N$,
large-$M$ limit, we obtain the saddle-point equations satisfied by the
collective fields, which determine the typical timescales of the dissipative
evolution, particularly, the spectral gap that rules the relaxation of the
system to its steady state. We solve the saddle-point equations numerically and
find that, for strong or intermediate dissipation, the system relaxes
exponentially, with a spectral gap that can be computed analytically, while for
weak dissipation, there are oscillatory corrections to the exponential
relaxation. In this letter, we illustrate the feasibility of analytical
calculations in strongly correlated dissipative quantum matter.
