Quasinormal Modes and the Hawking-Unruh Effect in Quantum Hall Systems: Lessons from Black Hole Phenomena

In this work, we propose the quantum Hall system as a platform for exploring black hole phenomena. By exhibiting deep rooted commonalities between the lowest Landau level and spacetime symmetries, we show that features of both quantum Hall and gravitational systems can be elegantly captured by a simple quantum mechanical model: the inverted harmonic oscillator. Through this correspondence, we argue that radiation phenomena in gravitational situations, such as presented by W. G. Unruh and S. Hawking, bear a parallel with saddle-potential scattering of quantum Hall quasiparticles. We also find that scattering by the quantum Hall saddle potential can mimic the signature quasinormal modes in black holes, such as theoretically demonstrated through Gaussian scattering off a Schwarzschild black hole by C. V. Vishveshwara. We propose a realistic quantum Hall point contact setup for probing these temporally decaying modes in quasiparticle tunneling, offering a new mesoscopic parallel for black hole ringdown.

Figure 1

(a) Schematic of the inverted harmonic oscillator potential and its scattering features of (i) regular transmission ($T$) and reflection ($R$) of an incident state ($I$); and (ii) temporally decaying quasinormal modes (QNMs), as applicable to two physically distinct phenomena [Figs. 1 and 1]. (b) Dynamics of a field in a black hole spacetime resulting in (i) Hawking-Unruh radiation, and (ii) signature QNMs associated with black hole ringdown. (c) Scattering in the quantum Hall lowest Landau level via a saddle potential leads to (i) quasiparticle tunneling; and (ii) as we predict, QNMs that could be observed for dilute enough quasiparticle sources.

Figure 2

Plots of wave-packet scattering off the IHO showing quasinormal mode behavior, as obtained form analytical calculations. (a) Pole structure of the $S$ matrix of the IHO in the complex energy plane. Blue crosses indicate the resonant poles of the IHO, and purple boxes indicate generic resonance poles for some arbitrary potential barrier. Closing the contour in the lower half-plane is determined by the fact that we have picked outgoing boundary conditions. (b) A wave packet composed of scattering energy eigenstates, impinging on the barrier from the right. (c) The scattered wave packet shows the “quasinormal ringdown“; the form takes into account only a single pole for illustrative purposes. The scattered state escapes to infinity, as seen in its finite amplitude at large $x$ but, as shown in Fig. 2, it exhibits an exponential time decay for a given point $x$ after $t>\tau \log |x/\ell |$.