Echo spectroscopy of Anderson localization

We propose a framework to study the onset of Anderson localization in disordered systems. The idea is to expose waves propagating in a random scattering environment to a sequence of short dephasing pulses. The system responds through coherence peaks forming at specific echo times, each echo representing a particular process of quantum interference. We suggest a concrete realization for cold gases, where quantum interferences are observed in the momentum distribution of matter waves in a laser speckle potential, and discuss in detail corresponding echoes in momentum space for sequences of one and two dephasing pulses. Our proposal defines a challenging but arguably realistic framework promising to yield unprecedented insight into the mechanisms of Anderson localization.

Figure 1

Physical observables represented in terms of pairs of retarded (solid lines) and advanced (dashed lines) Feynman path amplitudes. (a) Copropagating Feynman paths, $\alpha =\beta$, yield the classical contribution to the two-point transition probability $\mathbf{r}\to {\mathbf{r}}^{\prime }$. Inset: Weak-localization loop. (b) Coherent contribution, $\beta =\mathrm{T}\alpha$, to return probability $\mathbf{r}\to \mathbf{r}$, where $\mathrm{T}\alpha$ is the time reverse of $\alpha$. (c) Coherent backscattering contribution in the presence of dephasing pulses (wiggly lines). While a pulse at time $t_1$ (dashed wiggly lines) suppresses the phase coherence of generic loops, it affects particle and hole amplitudes in synchronicity if the loop is traversed in time ${\tau }_1=2t_2$, where coherence is briefly restored. Right: Synchronicity condition for a bitemporal pulse at times $t_{1,2}$ is realized at traversal time ${\tau }_2=t_1+t_2$, where a coherence signal is observed.

Figure 2

Coupling of diffuson (D) and Cooperon (C) to an external field $A=(\varphi ,\mathbf{a})$. The field acts at position $\mathbf{r}$ at the passage time of particle (solid upper line) and hole (dashed lower line), respectively. If these positions differ, dephasing occurs. In panel C, $\mathrm{T}A=\mathrm{T}(\varphi ,\mathbf{a})=(\varphi ,-\mathbf{a})$ indicates time reversal.

Figure 3

Higher-order coherence contributions to the return amplitude probed by bitemporal pulsing. The dephasing vertices shown in the inset of the right panel are only present in the C2b process. For the definition of the observation times ${\tau }_{3,4}$ and further discussion, see text.

Figure 4

Chronology of quantum coherence echoes in the $k_x\text{−}{k}_y$ plane of momentum space. Echoes are indicated by red dots whose width/position hint at the signal strength/angular orientation on the elastic scattering manifold. From left to right: An initial state with well defined momentum yields the single-Cooperon (C1) backscattering peak after the transport time $\tau$. A first dephasing pulse at $t_1$ suppresses the C1 signal, which reappears at the first echo time ${\tau }_1=2t_1$. A second pulse at $t_2$ generates the bipulse C1 echo at ${\tau }_2=t_1+t_2$. Two-mode echoes appear in the forward scattering direction at ${\tau }_3=2(t_2-t_1)$ (D2, C2a) and ${\tau }_4=2t_2-t_1$ (C2b).

Figure 5

Single-mode Cooperon echo contrast $\delta X_{\mathrm{C}1}(t,\overline{\mathbf{q}})/X_0$, given by (11), as function of time $t$ and momentum $\overline{q}$ (in units of ${\tau }_{\text{e}}$ and $\Delta p$ and for $\overline{\mathbf{q}}\parallel \Delta \mathbf{p})$, with a dephasing pulse at $t_1=10{\tau }_{\text{e}}$. The momentum kick $\Delta \mathbf{p}$ initially displaces the entire momentum distribution, then dephasing sets in, and the signal only revives at the echo time ${\tau }_1=2t_1$. The echo contrast at exact backscattering $\overline{q}=0$ is shown as the black curve in the side panel. The real-space signal (6) follows after integration over $\overline{\mathbf{q}}$.

Figure 6

Two-mode diffuson signal $\delta X_{\mathrm{D}2}(t,0)$ in the forward direction relative to the nonpulsed contribution, i.e., half of (A2), as a function of $t/{\tau }_{\text{e}}$, after two dephasing pulses at $t_1=10{\tau }_{\text{e}}$ and $t_2=40{\tau }_{\text{e}}$. The black line shows the result of (A3), a pronounced echo around ${\tau }_3=60{\tau }_{\text{e}}$. The red dashed curve shows the analytical approximation, Eq. (A9), above the background.

Figure 7

Two-mode echo contrast, Eq. (36), relative to the nonpulsed signal (A2) as function of $\mathbf{q}\parallel \Delta \mathbf{p}$ and $t$ (in units of $\Delta p$ and ${\tau }_{\text{e}})$ after two dephasing pulses at $t_1=10{\tau }_{\text{e}}$ and $t_2=40{\tau }_{\text{e}}$. The principal echo of processes D2 and C2a appears in the exact forward direction at time ${\tau }_3=2(t_2-t_1)$, whereas the side echo C2b, Eq. (A16), appears shifted by $\Delta \mathbf{p}$ and at later time ${\tau }_4=2t_2-t_1$.