Decoherent time-dependent transport beyond the Landauer-Buttiker formulation: a quantum-drift alternative to quantum jumps

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 $leftvert 0rightrangle $ that undergoes decoherence. We show the equivalence with the decoherent steady state transport in presence of a B{u}ttikers 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 $leftvert 0rightrangle $ $equiv $ $ leftvert uparrow downarrow rightrangle $ and $leftvert 1rightrangle equiv $ $leftvert downarrow uparrow rightrangle $. 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 $leftvert 0rightrangle $ and $leftvert 1right 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.