Decohering d -dimensional quantum resistance

The Landauer scattering approach to four-probe resistance is revisited for the case of a d -dimensional disordered resistor in the presence of decoherence. Our treatment is based on an invariant-embedding equation for the evolution of the coherent reflection amplitude coefficient in the length of a one-dimensional disordered conductor, where decoherence is introduced at par with the disorder through an outcoupling, or stochastic absorption, of the wave amplitude into side (transverse) channels, and its subsequent incoherent reinjection into the conductor. This is essentially in the spirit of Büttiker’s reservoir-induced decoherence. The resulting evolution equation for the probability density of the four-probe resistance in the presence of decoherence is then generalized from the one-dimensional to the d -dimensional case following an anisotropic Migdal-Kadanoff-type procedure and analyzed. The anisotropy, namely, that the disorder evolves in one arbitrarily chosen direction only, is the main approximation here that makes the analytical treatment possible. A qualitative result is that arbitrarily small decoherence reduces the localization-delocalization transition to a crossover making resistance moments of all orders finite.