Auxiliary Observable Variational Method to Non-Steady Dissipative Dynamics in Fermionic Superfluids

While most open quantum systems admit steady-state solutions, specific systems---such as purely dissipative systems and those exhibiting dynamical limit cycles---defy such descriptions. We present a novel generic variational principle that enables the study of non-steady dynamics in these systems under various approximations. Our approach is based on an action that simultaneously governs the Lindblad and adjoint equations. To demonstrate its utility, we apply this framework to investigate fermionic superfluids subject to one- or two-body losses, employing the Hartree-Fock-Bogoliubov approximation. We derive analytical solutions for homogeneous systems in the BCS limit revealing a universal behavior: the superfluid order parameter invariably decays to zero at a finite, well-defined critical time. This work provides a powerful tool for exploring the rich, time-dependent phenomena in open quantum many-body systems beyond steady-state paradigms.