Spectral properties of dissipative chaotic quantum maps

Area of Science:

• Quantum chaos
• Statistical mechanics
• Semiclassical analysis

Background:

• Chaotic quantum systems exhibit complex spectral properties.
• Dissipation introduces unique dynamics into quantum systems.
• Semiclassical methods bridge quantum and classical descriptions.

Purpose of the Study:

• To investigate the spectral properties of a dissipative chaotic quantum map.
• To compare quantum propagator traces with classical Frobenius-Perron operator traces.
• To validate a new semiclassical trace formula.

Main Methods:

• Utilizing a recently discovered semiclassical trace formula.
• Analyzing the propagator of the reduced density matrix.
• Comparing with the classical Frobenius-Perron operator.
• Numerical simulations for finite Planck's constant.

Main Results:

• Traces of quantum and classical operators are identical in the semiclassical limit (Planck's constant -> 0).
• Good numerical agreement is observed for finite Planck's constant, especially near strange attractors.
• Agreement is strong when classical dynamics is dominated by a point attractor.

Conclusions:

• The semiclassical trace formula accurately describes dissipative chaotic quantum maps.
• Quantum and classical dynamics show remarkable correspondence under dissipation.
• The findings are particularly relevant for systems with strange or point attractors.