Characterization of hybrid quantum eigenstates in systems with mixed classical phase space

Area of Science:

• Quantum mechanics
• Statistical mechanics
• Complex systems

Background:

• Generic low-dimensional Hamiltonian systems exhibit structured, mixed classical phase space.
• Traditional Percival classification of quantum spectra into regular and irregular states is insufficient.
• Berry's conjecture and eigenstate thermalization hypothesis (ETH) are not universally applicable.

Purpose of the Study:

• To demonstrate the limitations of existing spectral classification in Hamiltonian systems.
• To explore quantum effects in a prototype Bose-Hubbard model.
• To highlight the hybridization of chaotic and regular regions in phase space.

Main Methods:

• Analysis of a prototype Bose-Hubbard model.
• Investigation of quantum spectral properties.
• Examination of ergodicity and localization phenomena.

Main Results:

• Quantum spectral classification beyond Percival's is necessary.
• Quantum effects like tunneling, scarring, and localization do not follow standard paradigms.
• The Bose-Hubbard model exhibits a complex interplay between chaotic and regular dynamics.

Conclusions:

• The Hilbert space richness of Hamiltonian systems surpasses traditional spectral classifications.
• Quantum phenomena in mixed phase spaces require new theoretical frameworks.
• Understanding ergodicity and localization is key to characterizing complex quantum systems.