Observation of 'scarred' wavefunctions in a quantum well with chaotic electron dynamics

Abstract

QUALITATIVE insight into the properties of a quantum-mechanical system can be gained from the study of the relationship between the system's classical newtonian dynamics, and its quantum dynamics as described by the Schrödinger equation. The Bohr–Sommerfeld quantization scheme—which underlies the historically important Bohr model for hydrogen-like atoms—describes the relationship between the classical and quantum-mechanical regimes, but only for systems with stable, periodic or quasi-periodic orbits¹. Only recently has progress been made in understanding the quantization of systems that exhibit non-periodic, chaotic motion. The spectra of quantized energy levels for such systems are irregular, and show fluctuations associated with unstable periodic orbits of the corresponding classical system^1–3. These orbits appear as 'scars'—concentrations of probability amplitude—in the wavefunction of the system⁴. Although wavefunction scarring has been the subject of extensive theoretical investigation^5–10, it has not hitherto been observed experimentally in a quantum system. Here we use tunnel-current spectroscopy to map the quantum-mechanical energy levels of an electron confined in a semiconductor quantum well in a high magnetic field^10–13. We find clear experimental evidence for wavefunction scarring, in full agreement with theoretical predictions¹⁰.