  We present a model for decoherence in time-dependent transport. It boils down
into a form of wave function that undergoes a smooth stochastic drift of the
phase in a local basis, the Quantum Drift (QD) model. This drift is nothing
else but a local energy fluctuation. Unlike Quantum Jumps (QJ) models, no jumps
are present in the density as the evolution is unitary. As a first application,
we address the transport through a resonant state $\left\vert 0\right\rangle $
that undergoes decoherence. We show the equivalence with the decoherent steady
state transport in presence of a B\"{u}ttiker's voltage probe. In order to test
the dynamics, we consider two many-spin systems whith a local energy
fluctuation. A two-spin system is reduced to a two level system (TLS) that
oscillates among $\left\vert 0\right\rangle $ $\equiv $ $ \left\vert \uparrow
\downarrow \right\rangle $ and $\left\vert 1\right\rangle \equiv $ $\left\vert
\downarrow \uparrow \right\rangle $. We show that QD model recovers not only
the exponential damping of the oscillations in the low perturbation regime, but
also the non-trivial bifurcation of the damping rates at a critical point, i.e.
the quantum dynamical phase transition. We also address the spin-wave like
dynamics of local polarization in a spin chain. The QD average solution has
about half the dispersion respect to the mean dynamics than QJ. By evaluating
the Loschmidt Echo (LE), we find that the pure states $\left\vert
0\right\rangle $ and $\left\vert 1\right \rangle $ are quite robust against the
local decoherence. In contrast, the LE, and hence coherence, decays faster when
the system is in a superposition state. Because its simple implementation, the
method is well suited to assess decoherent transport problems as well as to
include decoherence in both one-body and many-body dynamics.
