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. As an example application, we show analytically that in a discrete-time quantum walk, 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.

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• Anderson localization
• Localization
• Quantum walks
• Disordered systems