Observation of Infinite-Range Intensity Correlations above, at, and below the Mobility Edges of the 3D Anderson Localization Transition

We investigate long-range intensity correlations on both sides of the Anderson transition of classical waves in a three-dimensional disordered material. Our ultrasonic experiments are designed to unambiguously detect a recently predicted infinite-range $C_0$ contribution, due to local density of states fluctuations near the source. We find that these $C_0$ correlations, in addition to $C_2$ and $C_3$ contributions, are significantly enhanced near mobility edges. Separate measurements of the inverse participation ratio reveal a link between $C_0$ and the anomalous dimension ${\mathrm{\Delta }}_2$, implying that $C_0$ may also be used to explore the critical regime of the Anderson transition.

Figure 1

Spatial intensity correlations for the two types of experiments at 2.4 MHz. The scanned-source data show convincing evidence of infinite-range ($C_0$) correlations, which are suppressed when only a single source point is used. Lines are theoretical fits using the values of parameters given in Table 1. The inset shows the single-source data on a log-log scale in order to reveal the extent to which the expected $1/\mathrm{\Delta }r$ dependence is observed for $C_2$ and $C_3$ at intermediate length scales. Data have been averaged over a bandwidth of 750 kHz, and the error bars are the standard deviations associated with the data’s statistical fluctuations, which are observed to be inherently large near the Anderson transition. (The open symbols for the scanned-source data represent positions where the measurements are not as reliable because of a smaller signal-to-noise ratio.)

Figure 2

Spatial (a) and frequency (b) correlations measured near 1 MHz, showing the increase of long-range correlations near a mobility edge. The inset shows the characteristic $1/\sqrt{\mathrm{\Omega }}$ behavior expected for $C_2$ and $C_3$ correlations. For both plots, data have been averaged over a bandwidth of 25 kHz, except at the highest frequency (1.11 MHz), where the data are changing too rapidly with frequency to be meaningfully averaged. These rapid variations with frequency near 1.11 MHz also complicate measurements of frequency correlations, which are therefore not shown here. The error bars are calculated as in Fig. 1. Lines show the fits using the parameters given in Table 1.

Figure 3

(a) Frequency dependence of the asymptotic value of the spatial correlation $C_{\omega }(\mathrm{\Delta }r\to \infty )=C_0^{(\text{in})}$ for single and scanned point sources, (b) anomalous dimension ${\mathrm{\Delta }}_2$ of the generalized inverse participation ratio for $q=2$, and (c) the amplitude transmission coefficient. The arrows in (c) indicate the resonant frequencies of individual aluminum beads. $C_0^{(\text{in})}$ and ${\mathrm{\Delta }}_2$ show large increases in magnitude near the upper band edges, where we expect mobility edges to occur. The grey vertical line indicates the location of the ME near 1.1 MHz, estimated from transverse confinement measurements.