We introduce time-dependent variational principles to study the nonunitary dynamics of open quantum many-body systems, including dynamics described by the full Lindblad master equation, the non-Hermitian dynamics corresponding to the no-click limit of the fully postselected quantum trajectories, and the dynamics described by a hybrid Lindbladian with a control parameter $\alpha$ which interpolates between the full postselection $(\alpha =0)$ and averaging over all quantum trajectories $(\alpha =1)$. As an application we study the nonunitary dynamics of a lossy or driven-dissipative BCS superconductors, evolving in the presence of two-body losses and two-body pumps. We show that the dynamics in the non-Hermitian limit is qualitatively different than the hybrid dissipative dynamics for any $\alpha >0$, leading to sharp modifications in the universal approach to the driven-dissipative steady states. By considering the dissipative dynamics with pair losses, we show that, as the non-Hermitian limit is approached, the density dynamics evolves from a universal power law to exponential decay that converges towards a quasisteady plateau characterized by the freezing of the particle depletion due to pair losses. The reached quasistationary density increases as a function of the dissipation rate highlighting the emergence of a non-Hermitian Zeno effect in the lossy dynamics. For the driven-dissipative case, the non-Hermitian dynamics skips the infinite temperature steady state reached in the presence of finite contribution of the quantum jumps, and the system can get trapped into an effective negative temperature state. We rationalize these findings in terms of the conservation of the length of the pseudospins which, in the non-Hermitian limit, suppresses the effective single-particle losses and pumps acting on the noncondensed particles.

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• Dynamical phase transitions
• Non-Hermitian systems
• Strongly correlated systems
• BCS theory
• Lindblad equation
• Variational approach