Ergodicity of complex dynamics and quantum tunneling in nonintegrable systems

Area of Science:

• Quantum mechanics
• Chaos theory
• Mathematical physics

Background:

• The instanton approximation is a standard semiclassical method for quantum tunneling.
• A key limitation of the instanton method is its requirement for energy-degenerate regions, hindering application to nonintegrable systems.

Purpose of the Study:

• To introduce a novel semiclassical approach for quantum tunneling in nonintegrable systems.
• To demonstrate how the ergodicity of complex orbits in Julia sets can overcome the limitations of the instanton approximation.

Main Methods:

• Utilizing the ergodicity of complex orbits within Julia sets to bridge energy regions.
• Applying semiclassical analysis in the time domain to investigate tunneling phenomena.
• Examining an ultra-near integrable system to isolate and study non-trivial tunneling behaviors.

Main Results:

• Ergodicity of Julia set complex orbits provides a connection between arbitrary energy regions, serving as an alternative to the instanton path.
• Nonmonotonic tunneling tails were observed in wave functions of an ultra-near integrable system, despite the absence of classical nonintegrability structures.
• Both the real and imaginary parts of the classical action were found to contribute to the step structure of tunneling tails, arising from quantum resonance.

Conclusions:

• The ergodicity of complex orbits in Julia sets offers a powerful new framework for semiclassical tunneling in nonintegrable systems.
• This method expands the applicability of semiclassical theories to complex dynamical systems.
• The study highlights the significant role of both real and imaginary components of classical action in quantum tunneling phenomena.