Sensitivity of quantum motion to perturbation in a triangle map

Area of Science:

• Quantum chaos
• Quantum dynamics
• Statistical mechanics

Background:

• The Loschmidt echo quantifies the sensitivity of quantum systems to perturbations.
• The triangle map serves as a model for studying quantum chaos due to its classical linear instability.
• Understanding fidelity decay is crucial for quantum information processing and quantum computing.

Purpose of the Study:

• To investigate the quantum Loschmidt echo (fidelity) in the triangle map.
• To identify and characterize different regimes of fidelity decay with respect to perturbation strength.
• To compare numerical findings with theoretical predictions for classical fidelity.

Main Methods:

• Numerical simulations of the quantum Loschmidt echo.
• Analysis of fidelity decay rates across different perturbation strengths.
• Comparison of quantum fidelity decay with classical fidelity predictions.

Main Results:

• Three distinct regimes of fidelity decay were observed: weak, intermediate, and strong perturbation.
• In weak perturbation, fidelity decays as exp(-c epsilon(2)t(gamma)) with gamma ≈ 1.7.
• In strong perturbation, fidelity follows a power-law decay, consistent with classical predictions, and decays slower than power-law for long times. An intermediate regime exhibits exponential decay exp(-c' epsilont).

Conclusions:

• The quantum Loschmidt echo in the triangle map exhibits complex behavior dependent on perturbation strength.
• Numerical results align with theoretical predictions for classical fidelity in the strong perturbation regime.
• The study highlights the intricate relationship between classical chaos and quantum fidelity decay.