  Yin and Radzihovsky [1] recently developed a self-consistent extension of a
Bogoliubov theory, in which the condensate number density $n_c$ is treated as a
mean field that changes with time, in order to analyze a JILA experiment by
Makotyn et al. [2] on a $^{85}$Rb Bose gas following a deep quench to a large
scattering length. We apply this theory to construct a closed set of equations
that highlight the role of $\dot{n}_c$, which is to induce an effective
interaction between quasiparticles. We show analytically that such a system
supports a steady state characterized by a constant condensate density and a
steady but periodically changing momentum distribution, whose time average is
described exactly by the generalized Gibbs ensemble. We discuss how the
$\dot{n}_{c}$-induced effective interaction, which cannot be ignored on the
grounds of the adiabatic approximation for modes near the gapless Goldstone
mode, can significantly affect condensate populations and Tan's contact for a
Bose gas that has undergone a deep quench.
