Probing Localization in Absorbing Systems via Loschmidt Echos

We measure Anderson localization in quasi-one-dimensional waveguides in the presence of absorption by analyzing the echo dynamics due to small perturbations. We specifically show that the inverse participation number of localized modes dictates the decay of the Loschmidt echo, differing from the Gaussian decay expected for diffusive or chaotic systems. Our theory, based on a random matrix modeling, agrees perfectly with scattering echo measurements on a quasi-one-dimensional microwave cavity filled with randomly distributed scatterers.

Figure 1

Left: Experimental setup—the top plate with two mounted antennae has been removed to show the waveguide with scatterers. Right: The variance of the normalized transmission intensity $\tilde{T}=|S_{21}{|}^2/⟨|S_{21}{|}^2⟩$. Vertical dashed lines are the mode cutoff frequencies. The numbers between are the number of open modes. The horizontal dashed line is the localization threshold of $7/3$. The shaded region is a localized frequency window of 6.0–7.5 GHz. Inset: Transmission distribution $\mathcal{P}(\tilde{T})$ for the localized frequency window of 6.0–7.5 GHz. The red dashed line is a fit (${\chi }^2=0.094$) of the core to a log-normal distribution, with a width of ${\sigma }_{\tilde{T}}^2=3.37$.

Figure 2

Experimental scattering fidelity in the frequency window of 6 to 7.5 GHz (localized). The dots correspond to the experimental scattering fidelity, Eq. (3), for a wall shift of 0.8 mm. The dashed line is a best fit to Eq. (4), while the solid line is a best fit to Eq. (2). Both fits were done for the region $t\le t_H$. Equation (2) can be seen as the better fit.

Figure 3

Fidelity, Eq. (1), for Hamiltonians modeled by BRM. The parameters are $\lambda ={10}^{-3}$, $L=5000$, $b=10$ (upper curve), and $b=3$ (lower curve). Circles are the numerical results. Dashed lines are best fits to Eq. (4). Solid lines are best fits to Eq. (2), which again show a better fit. The inset shows the variance extracted from the fit of Eq. (2), plotted against the root of the inverse participation number. The observed linear relation confirms the validity of Eq. (2).

Figure 4

Within the localized frequency window of 6–7.5 GHz, the experimental fitting parameters—$\alpha$ (squares) from Eq. (2) and $\lambda$ (circles) from Eq. (4)—are plotted against the rescaled wall shift $w/\delta w$. Straight lines indicate best fit to power laws, $\alpha \sim w^{0.92}$, $\lambda \sim w^{1.9}$. Inset: Same as main figure, but for the delocalized frequency window of 10.5–12 GHz. As opposed to the main figure, here $\lambda \sim w^{1.0}$, in agreement with Eq. (4).