Generating Particlelike Scattering States in Wave Transport

We introduce a procedure to generate scattering states which display trajectorylike wave function patterns in wave transport through complex scatterers. These deterministic scattering states feature the dual property of being eigenstates to the Wigner-Smith time-delay matrix $Q$ and to the transmission matrix $t^{†}t$ with classical (noiseless) transmission eigenvalues close to 0 or 1. Our procedure to create such beamlike states is based solely on the scattering matrix and successfully tested numerically for regular, chaotic, and disordered cavities. These results pave the way for the experimental realization of highly collimated wave fronts in transport through complex media with possible applications such as secure and low-power communication.

Figure 1

Scattering through a rectangular cavity (flux injected through the left lead of width $d$): (a),(b) Wave function densities of transmission eigenstates $|\tau ⟩$ of $t^{†}t$ with similar transmission eigenvalues $\tau >0.99$ but different wave numbers: (a) $k=5.5\pi /d$, (b) $k=75.5\pi /d$. (c) Classical surface of section, recorded for trajectories which enter at the left lead mouth with vertical position $y$ and transverse momentum $p_y$. The largest of the transmission (reflection) bands [black (white)] are labeled by T1, T2 (R1, R2) (bands which are equivalent in an extended zone scheme are given the same label). (d) Husimi distributions of the states shown in (a) (upper panel) and in (b) (lower panel). The size (area) of the Planck cell $h$ is indicated by dashed black frames and the underlying classical phase space is shown in gray.

Figure 2

(a) Transmission $T$ vs null-space norm $\chi$ for eigenstates of the matrix $Q_{11}$ in a rectangular cavity with different lead orientations (green dots). NOTEs with $\chi \to 0$ are noiseless and strongly deviate from RMT (red diamonds, see 14 B for details). (b)–(d) Wave function densities for NOTEs calculated with the same scattering matrix data as used for Fig. 1b. As demonstrated by the Husimi plots in the bottom panels, each state is located on a single classical phase space band. Null-space projections $\chi$ are (b) 4.7, (c) 6.3, (d) 6.9. The insets in (c) illustrate the possibility to use NOTEs for transferring information between a sender ($A$) and a receiver ($B$) which bypasses a potential eavesdropper ($E$).

Figure 3

NOTEs at $k=75.5\pi /d$ in cavities (a),(b) with a Sinai-type boundary shape and (c),(d) with a bulk disorder of amplitude $V_0/E=0.1$ (see illustrations). The correlation of the disorder is long-range for (c) with correlation length $k{r}_c=30\pi$ and short-range for (d) where $k{r}_c=5\pi$. Null-space norms $\chi$ are (a) 1.7, (b) 24.8, (c) 4.9, (d) 65.7.