Revival resonant scattering, perfect caustics, and isotropic transport of pseudospin-1 particles

We report unusual physics associated with wave scattering in pseudospin-1 systems whose band structure consists of a conventional Dirac cone and a topologically flat band. First, for small scatterer size, we find a surprising revival resonant scattering phenomenon and identify a peculiar type of boundary trapping profile through the formation of unusual vortices as the physical mechanism. Second, for larger scatterer size, a perfect caustic phenomenon arises as a manifestation of the super-Klein tunneling effect, leading to the scatterer's being effectively as a Veselago lens. Third, in the far scattering field, an unexpected isotropic behavior emerges at low energies, which can be attributed to the vanishing Berry phase for massless pseudospin-1 particles and, consequently, to constructive interference between the time-reversed backscattering paths. We develop an analytic theory based on the generalized Dirac-Weyl equation to fully explain these phenomena and articulate experimental schemes with photonic or electronic systems.

Figure 1

General behaviors of the scattering efficiency versus the scatterer strength. Scattering efficiency $Q$ as a function of the scatterer strength $V_{0}R$ for a number of representative values of the relative incident energy: (a) $E/V_0=0.01$, (b) $E/V_0=0.1$, and (c) $E/V_0=0.9$. The middle and bottom panels show the corresponding probability density and local current density profiles, respectively. In (a)–(c), the blue curves are for the conventional massless pseudospin/spin-$1/2$ case, while the red ones are for the massless pseudospin-1 wave system. Certain features can be understood by referring to the resonance width expressions ${\mathrm{\Gamma }}_1\propto {\left(kR\right)}^3$ for the resonances associated with the $\left|l\right|=1$ channel and ${\mathrm{\Gamma }}_0\propto \left(kR\right)\gg {\mathrm{\Gamma }}_1$ for those with the $l=0$ channel. For example, in (a), the second and fourth modes belong to the normal excitations of the $\left|l\right|=1$ channel, which are located about the first ($V_{0}R\sim 3.8$) and the second ($V_{0}R\sim 7$) zeros of the Bessel function $J_1$ in the domain of $V_{0}R$, while the first, third, and fifth modes are the excitations of normal modes belonging to the $l=0$ channel, which are located about the first ($\sim 2.4$), the second ($\sim 5.5$), and the third ($\sim 8.7$) zeros of the Bessel function $J_0\left(V_{0}R\right)$. This is why the long lifetime (corresponding to narrow resonance width) appears in the fourth mode but not in the third and fifth modes. This argument holds in the regime $kR\ll 1$ and $E/V_0\ll 1$.

Figure 2

Scattering efficient $Q$ versus the scatterer strength $V_{0}R$ and the relative incident energy $E/V_0$ for (a) massless pseudospin-$1/2$ and (b) massless pseudospin-1 wave systems. The black curve is for $E/V_0=0.49$ with a highlighted visual effect.

Figure 3

Vortex-based wave trapping for massless pseudospin-1 scattering. (a)–(c) Schematic illustration of three types of vortex-based wave trapping. (d) Wave pattern near the boundary for a massless pseudospin-$1/2$ particle for $E/V_0=0.086$. (e), (f) Boundary wave patterns for the massless pseudospin-1 case for $E/V_0=0.044$ and $E/V_0=0.487$, respectively. The value of $V_{0}R$ is set to be 4.5 for all cases.

Figure 4

Comparison of conventional resonant, revival resonant, and whispering gallery modes for the pseudospin-1 Dirac cone system. (a)–(c) Typical probability distributions for the conventional resonant mode, revival resonant mode (RRM), and whispering gallery mode (WGM), respectively, where the scattering parameters $[E/V_0,V_{0}R]$ are (a) [0.2, 11.598], (b) [0.487,4.5], and (c) [0.4157, 100]. (d)–(f) The corresponding current patterns with the magnitude coded by colors: monotonically increasing from blue to red.

Figure 5

Caustic behavior in the semiclassical regime and perfect caustics in pseudospin-1 Dirac cone systems. The probability and local current density patterns, respectively, for [(a),(c)] conventional pseudo-$1/2$ and [(b),(d)] pseudospin-1 Dirac cone systems. The probability density patterns in (a) and (b) are plotted on a logarithmic scale. The corresponding local current density patterns in (c) and (d) are color coded with magnitude normalized by its maximum. The parameters are $kR=300$ and $E/V_0=1/2$.

Figure 6

Super-Klein tunneling in pseudospin-1 Dirac cone systems. Transmission probability: red and blue curves are for the massless pseudospin-1 and pseudospin-$1/2$ particles, respectively, with $E/V_0=0.5$ (solid lines) and $E/V_0=0.4$ (dashed lines). For the former (solid red line) the transmission is unity, regardless of the incident angle.

Figure 7

Robustness of super-Klein tunneling and perfect caustics in pseudospin-1 Dirac cone systems. Top panels: polar plots of the transmission probability for different values of $E/V_0$ as indicated. The left column is for the pseudospin-$1/2$ system, while the right is for the pseudospin-1 system. Parameters used for generating the caustic patterns are $kR=300$ and $E/V_0=0.4$ (rather than exactly at $E/V_0=0.5$).

Figure 8

Isotropic scattering of a massless pseudospin-1 quasiparticle. Ratio $\xi$ as a function of $kR$ for $V_{0}R=5$: the red and blue lines are for the massless pseudospin-1 and pseudospin-$1/2$ cases, respectively.