  For a coherent quantum mechanical two-level system driven with a linearly
time-dependent detuning, the Landau-Zener model has served over decades as a
textbook model of quantum dynamics. A particularly intriguing question is
whether that framework can be extended to capture an intrinsical nonequilibrium
nature for a quantum system with coherent and dissipative dynamics occurring on
an equal footing. In this work, we are motivated to investigate the
Landau-Zenner problem of polariton condensates in a periodic potential under
nonresonant pumping, considering driven-dissipative Gross-Pitaevskii equations
coupled to the rate equation of a reservoir. Using a two-mode approach, we find
fluctuation of the reservoir can be considered as a constant and the relative
phase plays a very important role. The evolution of the dissipative
Landau-Zener model we obtain presents its adiabatic process very different from
the closed system because the fluctuation of the reservoir has a peak and leads
to the damping of the condensates. We substitute the fluctuation of the
reservoir to Hamiltonian and get an effective two-level model. The motion of
Hamiltonian in phase space is also discussed and is directly corresponding to
the pumping rate. The instability of the band structure can also be studied by
the curvatures in phase space and there may be two loops in the middle of the
Brillouin zone when the pumping rate is far beyond the threshold.
