Scattering expansion for localization in one dimension: From disordered wires to quantum walks

We present a perturbative approach to disordered systems in one spatial dimension that accesses the full range of phase disorder and clarifies the connection between localization and phase information. We consider a long chain of identically disordered scatterers and expand in the reflection strength of any individual scatterer. We apply this expansion to several examples, including the Anderson model, a general class of periodic -on -averagerandom potentials, and a two -component discrete -time quantum walk, showing analytically in the latter case that the localization length can depend nonmonotonically on the strength of phase disorder (whereas expanding in weak disorder yields monotonic decrease). More generally, we obtain to all orders in the expansion a particular nonseparable form for the joint probability distribution of the transmission coefficient logarithm and reflection phase. Furthermore, we show that for weak local reflection strength, a version of the scaling theory of localization holds: the joint distribution is determined by just three parameters.