PHASE DISTRIBUTION IN 1D LOCALIZATION AND PHASE TRANSITIONS IN SINGLE-MODE WAVEGUIDES

Localization of electrons in 1D disordered systems is usually described in the random phase approximation, when distributions of phases j and q, entering the transfer matrix, are considered as uniform. In the general case, the random phase approximation is violated, and the evolution equations (when the system length L is increased) contain three independent variables, i.e. the Landauer resistance r and the combined phases y = q j and c = q + j. The phase c does not affect the evolution of r and was not considered in previous papers. The distribution of the phase y is found to exhibit an unusual phase transition at the point Ɛ₀ when changing the electron energy Ɛ, which manifests itself in the appearance of the imaginary part of y. The resistance distribution P(r) has no singularity at the point Ɛ₀, and the transition looks unobservable in the electron disordered systems. However, the theory of 1D localization is immediately applicable to propagation of waves in single-mode optical waveguides. The optical methods are more efficient and provide possibility to measure phases y and c. On the one hand, it makes observable the phase transition in the distribution P(y), which can be considered as a “trace” of the mobility edge remaining in 1D systems. On the other hand, observability of the phase c makes actual derivation of its evolution equation, which is presented below. Relaxation of the distribution P(c) to the limiting distribution P_∞ (c) at L → ∞ is described by two exponents, whose exponentials have jumps of the second derivative, when the energy Ɛ is changed.