  We study the dynamic response of an s-wave BCS-BEC (atomic-molecular)
condensate to detuning quenches within the two channel model beyond the weak
coupling BCS limit. At long times after the quench, the condensate ends up in
one of three main asymptotic states (nonequilibrium phases), which are
qualitatively similar to those in other fermionic condensates defined by a
global complex order parameter. In phase I the amplitude of the order parameter
vanishes as a power law, in phase II it goes to a nonzero constant, and in
phase III it oscillates persistently. We construct exact quench phase diagrams
that predict the asymptotic state (including the many-body wavefunction)
depending on the initial and final detunings and on the Feshbach resonance
width. Outside of the weak coupling regime, both the mechanism and the time
dependence of the relaxation of the amplitude of the order parameter in phases
I and II are modified. Also, quenches from arbitrarily weak initial to
sufficiently strong final coupling do not produce persistent oscillations in
contrast to the behavior in the BCS regime. The most remarkable feature of
coherent condensate dynamics in various fermion superfluids is an effective
reduction in the number of dynamic degrees of freedom as the evolution time
goes to infinity. As a result, the long time dynamics can be fully described in
terms of just a few new collective dynamical variables governed by the same
Hamiltonian only with "renormalized" parameters. Combining this feature with
the integrability of the underlying (e.g. the two channel) model, we develop
and consistently present a general method that explicitly obtains the exact
asymptotic state of the system.
