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Adiabatic quantization of Andreev quantum billiard levels.
P G Silvestrov1, M C Goorden, C W J Beenakker
1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands.
Physical Review Letters
|April 12, 2003
Summary
We found that the time between Andreev reflections is a classical adiabatic invariant in chaotic cavities. This leads to quantized energy levels and squeezed wave functions, revealing new quantum phenomena in superconducting systems.
Area of Science:
- Quantum physics
- Condensed matter physics
- Mesoscopic systems
Background:
- Andreev reflections are crucial for understanding electron transport in hybrid superconductor-normal metal systems.
- Chaotic cavities exhibit complex electron dynamics influenced by their geometry and coupling.
- Adiabatic invariants are fundamental concepts in classical and quantum mechanics, describing conserved quantities during slow transformations.
Purpose of the Study:
- To identify and characterize classical adiabatic invariants in ballistic chaotic cavities coupled to superconductors.
- To investigate the consequences of adiabatic invariance quantization on energy levels and wave function properties.
- To explore the relationship between the Ehrenfest time, system parameters, and quantum behavior.
Main Methods:
- Theoretical analysis of electron transport in a ballistic chaotic cavity coupled to a superconductor via an N-mode constriction.
- Identification of the time between Andreev reflections as a classical adiabatic invariant.
- Quantization of the adiabatic invariant torus in phase space to determine discrete periods and energy states.
- Projection of the invariant torus onto the coordinate plane to analyze wave function spatial distribution.
Main Results:
- The time T between Andreev reflections is identified as a classical adiabatic invariant.
- Quantization of the invariant torus yields discrete periods T(n) and a ladder of excited states epsilon (nm).
- The largest quantized period, the Ehrenfest time, is found to be T(0)=lambda(-1)ln(N).
- Wave functions inside the cavity are squeezed to a transverse dimension W/sqrt[N], significantly smaller than the constriction width W.
Conclusions:
- The study reveals a novel connection between classical adiabatic invariants and quantum phenomena in mesoscopic superconducting systems.
- Quantization of the adiabatic invariant leads to discrete energy spectra and spatially confined wave functions.
- The findings provide insights into the behavior of electrons in chaotic cavities and their interaction with superconductors, with implications for quantum device design.