Figure 1

Two reverberant cavities are weakly coupled through a small window. A single channel through which an ingoing pulse returns a diffuse reflection $s\left(t\right)$ is attached to one of the cavities.Reuse & Permissions

Figure 2

(Color online) (a) A comparison of the squared and smoothed incoherent Sabine reflection $⟨s^2\left(t\right)⟩$ (dashed line) and the repaired $S$ matrix $⟨S^2\left(t\right)⟩$ [paler (red online) solid line] for case $A$. The rooms are well coupled ($\eta =10$, $T_H=2000$) and behave together like one large room. A power law tail $⟨S^2\left(t\right)⟩\sim t^{-5∕2}$ is visible at late times. (b) The enhanced backscatter factor of 2 is seen in this closeup of the data of (a). The repair has reproduced the fluctuations in the original. (c) Case $A$ continued. The reverberant part of $[-{\partial }_{t}K(t)∕2]=\mathrm{reverberant}\mathrm{part}\mathrm{of}$ $\partial G^0∕dt$, i.e., the part without the delta function at time zero, $S∕(1+S)$, is squared and smoothed (dashed line) and compared with the Sabine-type prediction $\sigma F_1\left(t\right)$ (solid line). An enhanced backscatter factor of 2 is noticeable at early times. A quantum echo appears later in which the enhancement grows to three. As predicted in random matrix theory [19] the quantum echo arrives on a time scale comparable to the effective Heisenberg time, 4000. (d) A closeup of the short time behavior of (c) shows the enhanced backscatter factors. The Sabine $\sigma F_1$ (solid line) shows the leaking process as probability flows from the first room into the second on a time scale of $t_{\mathrm{leak}}=200$.Reuse & Permissions

Figure 3

(Color online) (a) A comparison of the squared and smoothed incoherent Sabine reflection $⟨s^2\left(t\right)⟩$ (dashed line) and the repaired $S$ matrix $⟨S^2\left(t\right)⟩$[paler (red online) solid line] for case $B$ for which the rooms are moderately coupled ($\eta =0.5$, $T_H=2000$). (b) The behavior of the Sabine (dashed line) and repaired recapture rates (solid line) for case $B$ at early time. As in Fig. 2b one observes enhanced backscatter and reproduction in $S$ of the fluctuations in $s$. (c) The Sabine $\sigma F_1$ for case $B$ (solid line) shows the relaxation of room one’s energy density on the scale of the leak time as energy flows into the second room. The repaired $S$ matrix corresponds to an energy density (dashed line) at the source with a very different behavior. At late times it is enhanced over the Sabine prediction by a factor of 4.4.Reuse & Permissions

Figure 4

(Color online) (a) The smoothed squares of the Sabine $s\left(t\right)$ (dashed line) and the repaired $S\left(t\right)$ [pale solid line (red online)] are compared for the case $C$, for which the cavities are weakly coupled. (b) The Sabine $\sigma F_1$ (solid line) for case $C$ shows the slow relaxation of room one’s energy on the scale of the leak time as energy flows into the second room. The repaired $S$ matrix corresponds to an energy density at the source with a very different behavior. At late times it is enhanced over the Sabine prediction by a factor of 5.2.Reuse & Permissions

Figure 5

A single channel is attached to the midpoint of a finite one-dimensional diffusing medium. An ingoing pulse reflects diffusely.Reuse & Permissions

Figure 6

(Color online) (a) The Sabine envelope $\surd \sigma E$ for a diffusing medium (case $D$, with $T=5000$, $\Xi =0.25$) is multiplied by white noise of unit root mean square, squared, smoothed, and plotted, $⟨s_t^2⟩$, in the dashed line. At late times it decays exponentially. The smoothed square of the repaired $S_t$ is plotted in the pale (red online) solid line. It decays at a late time like a power law $\sim t^{-5∕2}$. (b) Comparison of smoothed squared ${\partial }_{t}K\left(t\right)∕2$ (irregular line) with the Sabine $\sigma F(x=0,t)$ (smooth line) for the diffusing medium case $D$, with small $\Xi$. An early time enhancement by a factor of 2 is followed by a later time enhancement that grows to four, slightly greater than the usual quantum echo factor of 3. In accord with the small value of $\Xi$, localization is not strong. (The apparent increase in fluctuation strength at late times is not real, being an artifact of the increasing density of points in the plot.)Reuse & Permissions

Figure 7

(Color online) (a) The Sabine envelope $\surd \sigma E$ for a diffusing medium (case $E$, with $T=30000$, $\Xi =1.0$) is multiplied by white noise of unit root mean square, squared, smoothed, and plotted, $⟨s_t^2⟩$, in the dashed line. At late times it decays exponentially. The smoothed square of the repaired $S_t$ is plotted in the pale (red online) solid line. It decays at a late time like a power law $\sim t^{-5∕2}$. (b) Comparison of smoothed squared ${\partial }_{t}K\left(t\right)∕2$ (irregular line) with the Sabine $\sigma F(x=0,t)$ (smooth line) for the diffusing medium case $E$, with moderate $\Xi$. An early time enhancement by a factor of 2 is followed by a later time enhancement that grows to 8. This indicates, after considering the quantum echo, which accounts for a factor of 3, a degree of localization consistent with a moderate value of $\Xi$.Reuse & Permissions

Figure 8

(Color online) (a) The recapture rates for case $F$ with $T=100000$ and $\Xi =5.0$. Again $⟨S^2⟩$ [pale solid curve (red online)] exhibits an enhanced backscatter factor of 2 at early times. As in Figs. 6a, 6b, it also reproduces much of the fluctuations in the original $⟨s^2⟩$ and shows a power law tail. (b) In case $F$ with its larger value of $\Xi$ $(=5)$, $({\partial }_t{G}^0{)}^2$ (irregular line) shows an asymptotic enhancement by a factor of about 18 over the Sabine prediction (smooth curve).Reuse & Permissions

Figure 9

(Color online) (a) The recapture rates for case $G$ with $T=100000$ and $\Xi =10.0$. The Sabine $⟨s_t^2⟩$ is given by the dashed line; the repaired $⟨S_t^2⟩$ is given by the solid line. (b) At the largest value of $\Xi$ considered (case $G$, $\Xi =10$) the energy at late times in $({\partial }_t{G}^0{)}^2$ (irregular line) is about 35 times greater than that predicted by Sabine (smooth line).Reuse & Permissions