Microwave conductance in random waveguides in the crossover to Anderson localization and single parameter scaling

The nature of transport of electrons and classical waves in disordered systems depends upon the proximity to the Anderson localization transition between freely diffusing and localized waves. The suppression of average transport and the enhancement of relative fluctuations in conductance in one-dimensional samples with lengths greatly exceeding the localization length, $L\gg \xi$, are related in the single parameter scaling (SPS) theory of localization. However, the difficulty of producing an ensemble of statistically equivalent samples in which the electron wavefunction is temporally coherent has so-far precluded the experimental demonstration of SPS. Here we demonstrate SPS in random multichannel systems for the transmittance $T$ of microwave radiation, which is the analogue of the dimensionless conductance. We show that for $L\sim4\xi$ a single eigenvalue of the transmission matrix (TM) dominates transmission and the distribution of the $\ln T$ is Gaussian with a variance equal to the average of $-\ln T$, as conjectured by SPS. For samples in the crossover to localization, $L\sim\xi$, we find a one-sided distribution for $\ln T$. This anomalous distribution is explained in terms of a charge model for the eigenvalues of the transmission matrix $\tau$ in which the Coulomb interaction between charges mimics the repulsion between the eigenvalues of transmission matrix. We show in the localization limit that the joint distribution of $T$ and the effective number of transmission eigenvalues determines the probability distributions of intensity and total transmission for a single incident channel.